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A micromechanical bandpass filter with adjustable bandwidth and bidirectional control of centre frequency
Sensors and Actuators A 187 (2012) 10–15
Contents lists available at SciVerse ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
A micromechanical bandpass filter with adjustable bandwidth and bidirectional control of centre frequency M.S. Hajhashemi ∗ , Alborz Amini, Behraad Bahreyni Mechatronic Systems Engineering, Simon Fraser University, Surrey, BC, Canada
a r t i c l e
i n f o
Article history: Received 25 May 2012 Received in revised form 24 July 2012 Accepted 2 August 2012 Available online 21 August 2012 Keywords: Micromechanical filter Adjustable Coupling Center frequency Bandwidth Independent tuning
a b s t r a c t This paper introduces a tunable, narrow-band, micromechanical filter whose bandwidth and centre frequency can be adjusted independently. The filter is made of two micro-resonators that are electrostatically coupled using a middle electrode. A low coupling strength results in nearly constant bandwidth while one tunes the centre frequency bi-directionally by applying a DC voltage to the coupling electrode. On the other hand, the bandwidth of the filter is independently modified by applying axial stress to one of the resonators, without affecting the signal attenuation through the filter. Analytic and numerical models for the behaviour of the filter are also presented. Test devices with a centre frequency of about 300 kHz were fabricated in a standard micromachining process. Experimental results support the design principle and validity of the proposed models. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Tunable bandpass filters are needed in many applications. In particular, covering multiple communication bands with different characteristics demands versatile systems that can adapt to the various conditions. One possible solution is to have multiple RF paths and switch between them based on the requirements of the system. However, this approach comes at the expense of higher cost, larger circuit size, and higher power consumption [1]. Using a tunable system which can be modified based on the requirements significantly simplifies the system design while increasing its flexibility [2,3]. In particular to micromachined filters made from silicon, temperature dependence of the material properties presents another case where tuning capability will improve the application range of the filter. In most cases, it is desired to have some control over the bandwidth (BW) and centre frequency (fc ) of the designed filters to enable handling of multiple waveforms, reduce the thermal effects, and compensate for the manufacturing tolerances [4]. In particular, achieving multi-standard RF filters requires independent tuning of BW and fc as specified by the standards [5]. The design of bandpass filters based on micromechanical resonators has been the subject of intensive research over the past two decades [6–8]. Narrow bandwidth bandpass filters are often
∗ Corresponding author. Tel.: +1 778 689 6446. E-mail address: [email protected] (M.S. Hajhashemi). 0924-4247/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sna.2012.08.008
realized through electrostatic or mechanical coupling of similar resonators. Coupled microresonators also have numerous applications in the design of sensors and micromechanical filters [9]. Mechanical resonators can be coupled at nodal points, which have a near zero displacement in order to provide a weak coupling [10,11]. A major shortcoming of mechanical coupling of the resonators is the inability to tune the coupling between the resonators after the fabrication [12,13]. To circumvent this issue, many groups have employed electrically coupled micro-resonators [14]. Galayko et al. derived an analytical formula for the effective spring constants of an array of electrically coupled resonators with high impedance bias voltage on the tuning electrode [15]. An advantage of electrostatically tunable filters is that some tuning of fc or BW may be achieved by modifying the bias voltage of the resonators [16]. However, changing the bias voltage of the resonators also modifies the amount of signal attenuation through the filter in addition to affecting the BW [17]. The main challenges for design of tunable filters are the correlations that exist between BW, fc , and bandpass attenuation (i.e., insertion loss) of such filters [14,18,19]. In this paper, we introduce a design that allows for independent modification of the BW and fc of a micromechanical filter. The bandwidth is changed by inducing tensile stress on structures, and hence modifying its effective stiffness. An advantage of this method of tuning of BW is having the nearly constant transmission losses. On the other hand, fc is bi-directionally modified through changing the electrostatic coupling between the resonators.
M.S. Hajhashemi et al. / Sensors and Actuators A 187 (2012) 10–15
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the permittivity of free space, A is the area of electrodes and g is the initial electrostatic gap. Using Taylor’s series expansion and rearranging the similar terms gives: Meff y¨ + eff y˙ + Km y =
1 2 ε0 A 2 V 1 + y + ··· 2 g g2
(2)
Electrical stiffness Ke is defined as: Ke = −ε0 A
V2 g3
(3)
Equation of motion can now be rewritten as: Meff y¨ + eff y˙ + (Km + Ke )y = Fig. 1. Definition of filter parameters: fL and fH are the lower and higher corner frequencies, respectively; rPB is the passband ripple; fc is the center frequency, defined as the average of fL and fH ; and BW is filter bandwidth, defined as the difference between fH and fL .
2. Device structure Fig. 1 introduces some important characteristics of a bandpass filter including bandwidth (BW), upper (fH ) and lower (fL ) corner frequencies, centre frequency (fc ), and passband ripple (rPB ). Our proposed device structure is shown in Fig. 2, where two beam resonators are electrostatically coupled through a coupling beam. The axial stress on one of the resonators is modified by applying an electrostatic force using electrodes at the end of resonators. During the filter operation, both resonators are biased with a DC voltage VBias and the input signal Vin is applied to one of the resonators. A DC coupling voltage, VCoupling , is also applied to the coupling beam through a low-impedance source so that to adjust the electrical spring constant in both resonators by the same value. Another DC voltage, VStress , generates an electrostatic force (F) and produce an axial strain on one of the resonators, and therefore, modifying only the poles associated with that resonator. A simple mechanical model for a resonator is a lumped massspring system where the energy dissipation in the system is modelled by a damper. Equation of motion for a parallel plate electrostatic resonator with motions along y-axis is: Meff y¨ + eff y˙ + Km y =
1 2
ε0 A (g − y)2
V2
(1)
where Meff and Km are effective mass and mechanical stiffness of the system, eff is the damping coefficient, V is the difference between the voltages of electrodes (i.e., V = VBias − VCoupling ), ε0 is
1 ε0 A 2 V 2 g2
(4)
Since Ke is negative, applying a voltage to the coupling beam changes the effective stiffness (i.e., Km + Ke ), and results in decreasing the resonance frequency of the system. In a system of electrostatically coupled micro-resonators, changing the bias voltage of each resonator affects its resonance frequency, and therefore, can be used as a modulating parameter to tune fc . This tunability is one of the most important advantages of electrostatic coupling and can correct small fabrication errors, which is not possible with the mechanical coupling. When the resonators are coupled through a high-impedance node (e.g., through placing a charge on the coupling beam and removing the source), both the centre frequency and bandwidth of the filter are affected since the degree of coupling between the two resonators depends on the coupling voltage. To break apart the dependence of two filter parameters on a single control signal, the resonators in this work are coupled through low-impedance node such that only the centre frequency is affected by the voltage applied to the coupling beam. Adjustment of the filter bandwidth is then achieved through application of a stress on one of the resonators. Galayko et al. used three springs to model the effect of high impedance bias voltage on the coupling electrode [15]. Two of these springs represent electrical softening effect in two adjacent resonators and the third spring (Kc ) represents the coupling. The strength of these springs is a function of physical feedthrough capacitances between the electrodes and substrate as well as the dimension of the electrostatic gap: Ke1 ≈ Ke2 = Kc ≈
(VBias − VCoupling )2 g2
C
(VBias − VCoupling )2 C 2 Cp g2
Fig. 2. Schematic model for the coupled micro-resonators and tuning the bandwidth with stress induced by electrostatic force F.
(5)
(6)
12
M.S. Hajhashemi et al. / Sensors and Actuators A 187 (2012) 10–15 Table 1 Design parameters of the coupled DETF resonators.
Fig. 3. Stress tunable, electrostatically coupled DETF resonators fabricated in SOIMUMPs process.
Parameter
Symbol
Resonator length Resonator width Electrostatic gap Electrode length Structure thickness
L W g Le h
Value 300 m 10 m 2 m 200 m 25 m
where VBias and VCoupling are the bias voltage of the resonators and the coupling electrode, respectively, g is the gap between the resonators, C is the capacitance of this gap, and Cp is the parasitic capacitance. Test devices were fabricated from a 25 m thick, single crystal silicon layer in the SOIMuMPS process from MEMSCAP [20]. An optical image of the fabricated device is shown in Fig. 3 with main device parameters defined on the figure and summarized in Table 1. Note that double-ended tuning fork (DETF) resonators were used instead of simple beam resonators. While the system analysis remains essentially the same, the choice of DETF as the resonator for the filter function allows us to study two vibration modes per resonator as shown in Fig. 4 [18,19]. The in-phase vibration mode of a DETF (e.g., Fig. 4-a) in general has a lower mode frequency
Fig. 4. In-phase (a and b) and out-of-phase (c and d) modes of vibration for two coupled DETF resonators. Note that the addition of beam extensions to the DETFs has resulted in significant separation of the in-phase and out-of-phase mode frequencies.
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Fig. 5. The lumped mechanical model for coupled micro-resonators is shown to the left where Ke1 and Ke2 are electrostatic softening springs. The equivalent electrical model is shown to the right where C0 represents the overall effect of feedthrough capacitors.
than the out-of-phase mode (e.g., Fig. 4-c) due to the longer effective length of the beams for the in-phase mode. On the other hand, the larger anchor losses for the in-phase vibration mode of a DETF result in a lower quality factor for this mode compared to the out-of phase mode. In order to create a larger separation between the inand out-of phase modes, we added two beam extensions between the anchors and the main tuning fork resonators. While these beam extensions essentially remain stationary, and hence, do not affect the out-of-phase mode, they become part of the resonating structure for the in-phase mode. This interaction lowers the frequency of the in-phase mode and avoids having spurious passbands near the desired passband. Fig. 4 demonstrates the finite element (FE) simulation results of two coupled DETF where the contribution of the beam extensions to the separation of the modes can also be observed.
Fig. 6. Simulation results for bidirectional tuning of fc by applying a DC coupling voltage on the coupling electrode.
3. Tuning the centre frequency of the filter The centre frequency of the filter, fc , can be modified bidirectionally by changing VCoupling . Connecting a low impedance source to the coupling electrode reduces the amount of coupling between resonators and therefore it is possible to change the centre frequency of the filter without affecting the bandwidth. Since Cp is about 50 times larger than C on the test devices, the output signal is mainly affected by the electrical softening springs (Ke1 and Ke2 ) as shown in Fig. 5. In order to predict the biased resonance frequencies of the system, it is necessary to find the exact value of the effective mechanical stiffness (Km ) and unbiased mechanical resonance frequency of the system (f0 ). The unbiased resonance frequency of the in-phase vibration mode for DETF resonator can be estimated from: f0 =
1 2
Keff Meff
(7)
With the inclusion of electrical stiffness, the biased resonance frequency of the system can be calculated from:
f = f0
1+
Ke Km
(8)
Fig. 6 illustrates the simulation results for bidirectional control of fc when the coupling voltage is varied. The data was obtained for VBias = 110 V without stress on the resonators. The relatively large voltage needed here is due to the limit on the minimum gap distance (i.e., 2 m) in the fabrication technology and can be reduced significantly if devices with smaller gaps are fabricated. Simulations on the lumped model were carried in CoventorWare [21]. To illustrate the change in the location of the peaks, we set the damping coefficient in the simulation to a low value to have clear sharp peaks. However, such response is undesired for a bandpass filter and our fabricated devices do not exhibit this. As it can be seen in Fig. 6, changing the electrical spring stiffness of both resonators by the same value moves the resonance frequencies of
Fig. 7. The experimental setup for testing the coupled micro-resonators.
them together to the left or right depending on the value and sign of the coupling bias voltage (i.e., BW remains constant). Experiments were performed inside a vacuum chamber using a Rohde & Schwarz ZVB4 vector network analyzer (VNA) in a configuration as shown in Fig. 7. Fabricated devices were packaged in an 84 pin ceramic PGA package and placed inside a vacuum chamber at an operating pressure of about 1 Torr. A transimpedance amplifier with a gain of 100 k was used to amplify the current signal from the sense electrodes of the micro-resonators. Port 1 of the VNA was used to drive the resonators while the signal at the output of the amplifier was connected to Port 2. Experimental results for bi-directional tuning of fc through changing the coupling voltage (VCoupling ) are shown in Fig. 8 and are summarized in Table 2 (VBias = 110 V, VStress = 0). As can be seen, the filter exhibits low ripple (i.e., less than 1 dB) over its passband even though it is terminated to virtual ground at the input of the amplifier. This can be explained by noting the relatively low Q of the resonators (about 5000). On the other hand, due to the fabrication limitations, the nominal gap between the electrodes was about 2.5 m. These factors contributed to a motional resistance of about 550 k for each one of the DETFs, making proper termination challenging, especially since the signal loss through the filter is a function of various damping mechanisms. For a practical implementation of micromachined filters using electrostatic resonators, one needs to minimize the gap between the electrodes and try to
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Fig. 8. Experimental data for bidirectional tuning of fc by applying a DC coupling voltage on the coupling electrode.
Table 2 Effect of the coupling voltage on the experimental and analytical values of resonant frequencies. VCoupling (V)
VStress (V)
Change in BW (Hz)
Analytical
Experimental
302.987 303.281 303.563 303.830 304.090 304.335 304.568
302.993 303.289 303.569 303.833 304.081 304.329 304.561
0 60 85 110 135 160 185
4. Tuning filter bandwidth Changing the electrical stiffness gives a reasonable range of tunability in BW and fc of the coupled structures. However, both of these parameters change at the same time while the amplitude of the transmitted signal is also affected. On the other hand, small changes in axial stress on the resonators will not considerably affect the transmission magnitude [22]. Applying a tensile stress onto the structure increases the spring constant of the system. The applied axial force (F) modifies the spring stiffness (Kb ) of a Clamped–Clamped (CC) beam according to [23]: 4EI 3 L − 4 tanh(L/4)
(9)
where E is the Young’s modulus of elasticity, I is geometric moment of inertia, L is the length of the beam, and 2 = F/EI. Using Taylor’s series for small L, Eq. (9) can be simplified to:
L2 2 1+ 40
Analytical
Experimental
0 20 40 67 101 142 190
0 20 38 62 90 118 148
The resonance frequency of a CC beam with a mass attached to its centre is found from:
keep the environmental conditions as constant as possible so that proper termination of the filter is possible. For the unbiased system, it is not possible to measure the resonance frequencies of the in-phase vibration mode. In this mode, the anchor of the DETF also shows some compliance that is hard to be modelled. Moreover, fabrication tolerances also change the resonance frequency of the system. Therefore, experimental resonance frequencies of the biased system were substituted in (8) to estimate the unbiased resonance frequency. After calculating parameters f0 and Km , it is possible to analytically predict the resonance frequencies for the biased system. The small deviation in calculation of resonance frequency is related to the effect of coupling spring (KC ) that slightly changes the BW.
192EI Kb = L3
Table 3 Effect of VStress on bandwidth.
fc (kHz)
−15 −10 −5 0 +5 +10 +15
Kb =
Fig. 9. Tuning bandwidht through changing VStress .
(10)
1 f0 = 2
Kb Mp + 0.38 M
(11)
where Mp and M are the mass of the attached mass and beam respectively. The resonance frequency after applying the stress will be:
f = f0
1+
0.3L2 F EhW 3
(12)
where f0 is the resonance frequency of system at rest. Electrostatic actuators were employed using two symmetrical electrodes at the beam-ends (see Fig. 3) in order to apply an axial stress on the resonator beam. Using Taylor’s series again and substituting electrostatic force in (12) gives an analytical formula for change in resonance frequency as a function of applied voltage for generating the electrostatic force. Considering that each resonator in DETF sees half of the force, the change in bandwidth is found from:
BW = f0
0.075L2 EhW 3
ε0 A 2g 2
V2
(13)
where ε0 is the permittivity of air. Moreover, A, L, h and W are the area, length, height and width of electrodes, g is the gap and V is the difference in applied voltages between electrodes. Eq. (13) shows that the change in resonance frequency, and therefore filter BW, is linearly proportional to the applied axial force or quadratically to the applied voltage on the stress induction electrodes [22]. Fig. 9 illustrates the experimental results with VBias = 110 V, VCoupling = 0 for the change in BW as a function of VStress . For the particular device shown in Fig. 3, BW could be modified from 250 Hz to 400 Hz with less than 1 dB ripple (rPB ) in the passband of the filter. As expected, the signal attenuation is not significantly affected as the BW is changed. Table 3 compares the analytic and experimental results for BW adjustment using this technique. These results show that in low stress values, experimental results are very close to the analytical results from (13). One of
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it also makes it possible to properly terminate the filter, and hence, offering better passband characteristics. Acknowledgments The authors would like to acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), Western Economic Diversification (WD), and Simon Fraser University. Fabrication services were provided through CMC Microsystems. References
Fig. 10. Optimization of coupling between the microresonators for maximum BW with fixed VStress = 30 V.
the reasons for unmatched data for high voltages (high stress) could be ignoring the nonlinear mechanical effects. Another important factor that should be studied is sensitivity of this method for changing BW. In [9] we introduced a signal processing method for analysis of perturbations in arrays of coupled resonators. It can be shown that an optimum value exists for the coupling strength between resonators, which results in the highest sensitivity in changing the system poles to the inserted perturbation. Fig. 10 shows that by setting the bias voltages of the resonators independently, it is possible to modify the strength of coupling between them and hence getting the maximum change in BW for a given amount of stress. 5. Conclusions In this paper, a micromechanical filter design was proposed with independent tuning of its centre frequency and bandwidth. Since the principal idea of this filter is based on an array of weakly coupled micro-resonators, changing the coupling value modifies the centre frequency of the filter without affecting the separation between the resonance frequencies of the resonators. On the other hand, bandwidth of the system was changed by applying an axial stress on one of the resonators, which then modified its mechanical spring constant. One of the advantages of this method is the nearly constant transmission losses as the bandwidth is modified. We also presented the experimental results for an array of two electrically coupled DETF resonators, confirming that both filter parameters could be adjusted as the analytic models predicted. The filter performance can be improved significantly if it is fabricated in a process technology that is capable of reducing the electrostatic gaps. This not only reduces the losses through the resonators, but
[1] K. Kawai, H. Okazaki, S. Narahashi, Center frequency, bandwidth, and transfer function tunable bandpass filter using ring resonator and J-inverter, in: Proceedings of the 39th European Microwave Conference, Rome, Italy, 2009. [2] G. Rebeiz, K. Entesari, I. Reines, S.-J. Park, M. El-tanani, A. Grichener, A. Brown, Tuning in to RF MEMS, IEEE Microwave Magazine 10 (October (6)) (2009) 55–72. [3] K. Kawai, H. Okazaki, S. Narahashi, Tunable ring resonator filter for duplexer, in: Proceedings of the 38th European Microwave Conference, Amsterdam, The Netherlands, 2008. [4] H. Chandrahalim, S.A. Bhave, Digitally-tunable MEMS filter using mechanicallycoupled resonator array, in: IEEE 21st International Conference on Micro Electro Mechanical Systems, Tucson, AZ, USA, 2008. [5] C. Andriesei, L. Goras, On frequency and quality factor independent tuning possibilities for RF band-pass filters with simulated inductors, Romanian Journal of Information Science and Technology 11 (4) (2008) 367–382. [6] K. Wang, C.T.-C. Nguyen, High-order medium frequency micromechanical electronic filters, Journal of Microelectromechanical Systems 8 (1999) 534–557. [7] F.D. Bannon, J.R. Clark, C.T. Nguyen, High-Q HF microelectromechanical filters, IEEE Journal of Solid-State Circuits 35 (4) (2000). [8] R. Lanz, P. Muralt, Bandpass filters for 8 GHz using solidly mounted bulk acoustic wave resonators, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 52 (6) (2005). [9] M.S. HajHashemi, B. Bahreyni, A novel signal processing method for analysis of perturbations in arrays of coupled resonators, in: International Conference on Solid-State Sensors, Actuators and Microsystems, Transducers’ 11, Beijing, 2011. [10] A. Wong, C. Nguyen, Micromechanical mixer-filters (“mixlers”), IEEE/ASME Journal of Microelectromechanical Systems 13 (1) (2004) 100–112. [11] V. Chivukula, J. Rhoads, Microelectromechanical bandpass filters based on cyclic coupling architectures, Journal of Sound and Vibration 329 (20) (2010) 4313–4332. [12] K. Wang, F.D. Bannon III, J.R. Clark, C.T.-C. Nguyen, Q-enhancement of microelectromechanical filters via low velocity spring coupling, in: IEEE Ultrasonics Symposium, 1997. [13] P. Stephanou, G. Piazza, C. White, M. Wijesundara, A. Pisano, Mechanically coupled contour mode piezoelectric aluminum nitride MEMS filters, in: MEMS 2006 Istanbul. 19th IEEE International Conference on, Istanbul, 2006. [14] S. Pourkamali, F. Ayazi, Electrically coupled MEMS bandpass filters; part I: with coupling element, Sensors and Actuators A 122 (2005) 307–316. [15] D. Galayko, A. Kaiser, B. Legrand, L. Buchaillot, C. Combi, D. Collard, Coupledresonator micromechanical filters with voltage tunable bandpass characteristic in thick-film polysilicon technology, Sensors and Actuators A 126 (2006) 227–240. [16] K. Lee, Y. Cho, Frequency tuning of a laterally driven microresonator using an electrostatic comb array of linearly varied length, in: International Conference Solid-state Sens, 1997. [17] J. Kwon, J. Choi, K. Kim, J. Sim, J. Kim, J. Kim, Frequency tuning of nanowire resonator using electrostatic spring effect, IEEE Transactions on Magnetics 45 (5) (2009) 2332–2335. [18] J. Yan, A.A. Seshia, K.L. Phan, P.G. Steeneken, J.T.M. van, Narrow bandwidth single-resonator MEMS tuning fork, in: Frequency Control Symposium, 2007. [19] J.R. Clark, A.C. Wong, C.T.C. Nguyen, Parallel-resonator HF micromechanical bandpass filters, in: International Conference on Solid-State Sensors and Actuators, Chicago, 1997. [20] 2 2012. Available: http://www.memscap.com/en mumps.html. [21] 2 2010. Available: http://www.coventor.com/products/coventorware/. [22] J.E.Y. Lee, B. Bahreyni, A.A. Seshia, An axial strain modulated double-ended tuning fork electrometer, Sensors and Actuators A 128 (2008) 395–400. [23] B. Bahreyni, C. Shafai, A resonant micromachined magnetic field sensor, IEEE Sensors Journal 7 (9) (2007) 1326–1334.
Contents lists available at SciVerse ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
A micromechanical bandpass filter with adjustable bandwidth and bidirectional control of centre frequency M.S. Hajhashemi ∗ , Alborz Amini, Behraad Bahreyni Mechatronic Systems Engineering, Simon Fraser University, Surrey, BC, Canada
a r t i c l e
i n f o
Article history: Received 25 May 2012 Received in revised form 24 July 2012 Accepted 2 August 2012 Available online 21 August 2012 Keywords: Micromechanical filter Adjustable Coupling Center frequency Bandwidth Independent tuning
a b s t r a c t This paper introduces a tunable, narrow-band, micromechanical filter whose bandwidth and centre frequency can be adjusted independently. The filter is made of two micro-resonators that are electrostatically coupled using a middle electrode. A low coupling strength results in nearly constant bandwidth while one tunes the centre frequency bi-directionally by applying a DC voltage to the coupling electrode. On the other hand, the bandwidth of the filter is independently modified by applying axial stress to one of the resonators, without affecting the signal attenuation through the filter. Analytic and numerical models for the behaviour of the filter are also presented. Test devices with a centre frequency of about 300 kHz were fabricated in a standard micromachining process. Experimental results support the design principle and validity of the proposed models. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Tunable bandpass filters are needed in many applications. In particular, covering multiple communication bands with different characteristics demands versatile systems that can adapt to the various conditions. One possible solution is to have multiple RF paths and switch between them based on the requirements of the system. However, this approach comes at the expense of higher cost, larger circuit size, and higher power consumption [1]. Using a tunable system which can be modified based on the requirements significantly simplifies the system design while increasing its flexibility [2,3]. In particular to micromachined filters made from silicon, temperature dependence of the material properties presents another case where tuning capability will improve the application range of the filter. In most cases, it is desired to have some control over the bandwidth (BW) and centre frequency (fc ) of the designed filters to enable handling of multiple waveforms, reduce the thermal effects, and compensate for the manufacturing tolerances [4]. In particular, achieving multi-standard RF filters requires independent tuning of BW and fc as specified by the standards [5]. The design of bandpass filters based on micromechanical resonators has been the subject of intensive research over the past two decades [6–8]. Narrow bandwidth bandpass filters are often
∗ Corresponding author. Tel.: +1 778 689 6446. E-mail address: [email protected] (M.S. Hajhashemi). 0924-4247/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sna.2012.08.008
realized through electrostatic or mechanical coupling of similar resonators. Coupled microresonators also have numerous applications in the design of sensors and micromechanical filters [9]. Mechanical resonators can be coupled at nodal points, which have a near zero displacement in order to provide a weak coupling [10,11]. A major shortcoming of mechanical coupling of the resonators is the inability to tune the coupling between the resonators after the fabrication [12,13]. To circumvent this issue, many groups have employed electrically coupled micro-resonators [14]. Galayko et al. derived an analytical formula for the effective spring constants of an array of electrically coupled resonators with high impedance bias voltage on the tuning electrode [15]. An advantage of electrostatically tunable filters is that some tuning of fc or BW may be achieved by modifying the bias voltage of the resonators [16]. However, changing the bias voltage of the resonators also modifies the amount of signal attenuation through the filter in addition to affecting the BW [17]. The main challenges for design of tunable filters are the correlations that exist between BW, fc , and bandpass attenuation (i.e., insertion loss) of such filters [14,18,19]. In this paper, we introduce a design that allows for independent modification of the BW and fc of a micromechanical filter. The bandwidth is changed by inducing tensile stress on structures, and hence modifying its effective stiffness. An advantage of this method of tuning of BW is having the nearly constant transmission losses. On the other hand, fc is bi-directionally modified through changing the electrostatic coupling between the resonators.
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the permittivity of free space, A is the area of electrodes and g is the initial electrostatic gap. Using Taylor’s series expansion and rearranging the similar terms gives: Meff y¨ + eff y˙ + Km y =
1 2 ε0 A 2 V 1 + y + ··· 2 g g2
(2)
Electrical stiffness Ke is defined as: Ke = −ε0 A
V2 g3
(3)
Equation of motion can now be rewritten as: Meff y¨ + eff y˙ + (Km + Ke )y = Fig. 1. Definition of filter parameters: fL and fH are the lower and higher corner frequencies, respectively; rPB is the passband ripple; fc is the center frequency, defined as the average of fL and fH ; and BW is filter bandwidth, defined as the difference between fH and fL .
2. Device structure Fig. 1 introduces some important characteristics of a bandpass filter including bandwidth (BW), upper (fH ) and lower (fL ) corner frequencies, centre frequency (fc ), and passband ripple (rPB ). Our proposed device structure is shown in Fig. 2, where two beam resonators are electrostatically coupled through a coupling beam. The axial stress on one of the resonators is modified by applying an electrostatic force using electrodes at the end of resonators. During the filter operation, both resonators are biased with a DC voltage VBias and the input signal Vin is applied to one of the resonators. A DC coupling voltage, VCoupling , is also applied to the coupling beam through a low-impedance source so that to adjust the electrical spring constant in both resonators by the same value. Another DC voltage, VStress , generates an electrostatic force (F) and produce an axial strain on one of the resonators, and therefore, modifying only the poles associated with that resonator. A simple mechanical model for a resonator is a lumped massspring system where the energy dissipation in the system is modelled by a damper. Equation of motion for a parallel plate electrostatic resonator with motions along y-axis is: Meff y¨ + eff y˙ + Km y =
1 2
ε0 A (g − y)2
V2
(1)
where Meff and Km are effective mass and mechanical stiffness of the system, eff is the damping coefficient, V is the difference between the voltages of electrodes (i.e., V = VBias − VCoupling ), ε0 is
1 ε0 A 2 V 2 g2
(4)
Since Ke is negative, applying a voltage to the coupling beam changes the effective stiffness (i.e., Km + Ke ), and results in decreasing the resonance frequency of the system. In a system of electrostatically coupled micro-resonators, changing the bias voltage of each resonator affects its resonance frequency, and therefore, can be used as a modulating parameter to tune fc . This tunability is one of the most important advantages of electrostatic coupling and can correct small fabrication errors, which is not possible with the mechanical coupling. When the resonators are coupled through a high-impedance node (e.g., through placing a charge on the coupling beam and removing the source), both the centre frequency and bandwidth of the filter are affected since the degree of coupling between the two resonators depends on the coupling voltage. To break apart the dependence of two filter parameters on a single control signal, the resonators in this work are coupled through low-impedance node such that only the centre frequency is affected by the voltage applied to the coupling beam. Adjustment of the filter bandwidth is then achieved through application of a stress on one of the resonators. Galayko et al. used three springs to model the effect of high impedance bias voltage on the coupling electrode [15]. Two of these springs represent electrical softening effect in two adjacent resonators and the third spring (Kc ) represents the coupling. The strength of these springs is a function of physical feedthrough capacitances between the electrodes and substrate as well as the dimension of the electrostatic gap: Ke1 ≈ Ke2 = Kc ≈
(VBias − VCoupling )2 g2
C
(VBias − VCoupling )2 C 2 Cp g2
Fig. 2. Schematic model for the coupled micro-resonators and tuning the bandwidth with stress induced by electrostatic force F.
(5)
(6)
12
M.S. Hajhashemi et al. / Sensors and Actuators A 187 (2012) 10–15 Table 1 Design parameters of the coupled DETF resonators.
Fig. 3. Stress tunable, electrostatically coupled DETF resonators fabricated in SOIMUMPs process.
Parameter
Symbol
Resonator length Resonator width Electrostatic gap Electrode length Structure thickness
L W g Le h
Value 300 m 10 m 2 m 200 m 25 m
where VBias and VCoupling are the bias voltage of the resonators and the coupling electrode, respectively, g is the gap between the resonators, C is the capacitance of this gap, and Cp is the parasitic capacitance. Test devices were fabricated from a 25 m thick, single crystal silicon layer in the SOIMuMPS process from MEMSCAP [20]. An optical image of the fabricated device is shown in Fig. 3 with main device parameters defined on the figure and summarized in Table 1. Note that double-ended tuning fork (DETF) resonators were used instead of simple beam resonators. While the system analysis remains essentially the same, the choice of DETF as the resonator for the filter function allows us to study two vibration modes per resonator as shown in Fig. 4 [18,19]. The in-phase vibration mode of a DETF (e.g., Fig. 4-a) in general has a lower mode frequency
Fig. 4. In-phase (a and b) and out-of-phase (c and d) modes of vibration for two coupled DETF resonators. Note that the addition of beam extensions to the DETFs has resulted in significant separation of the in-phase and out-of-phase mode frequencies.
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Fig. 5. The lumped mechanical model for coupled micro-resonators is shown to the left where Ke1 and Ke2 are electrostatic softening springs. The equivalent electrical model is shown to the right where C0 represents the overall effect of feedthrough capacitors.
than the out-of-phase mode (e.g., Fig. 4-c) due to the longer effective length of the beams for the in-phase mode. On the other hand, the larger anchor losses for the in-phase vibration mode of a DETF result in a lower quality factor for this mode compared to the out-of phase mode. In order to create a larger separation between the inand out-of phase modes, we added two beam extensions between the anchors and the main tuning fork resonators. While these beam extensions essentially remain stationary, and hence, do not affect the out-of-phase mode, they become part of the resonating structure for the in-phase mode. This interaction lowers the frequency of the in-phase mode and avoids having spurious passbands near the desired passband. Fig. 4 demonstrates the finite element (FE) simulation results of two coupled DETF where the contribution of the beam extensions to the separation of the modes can also be observed.
Fig. 6. Simulation results for bidirectional tuning of fc by applying a DC coupling voltage on the coupling electrode.
3. Tuning the centre frequency of the filter The centre frequency of the filter, fc , can be modified bidirectionally by changing VCoupling . Connecting a low impedance source to the coupling electrode reduces the amount of coupling between resonators and therefore it is possible to change the centre frequency of the filter without affecting the bandwidth. Since Cp is about 50 times larger than C on the test devices, the output signal is mainly affected by the electrical softening springs (Ke1 and Ke2 ) as shown in Fig. 5. In order to predict the biased resonance frequencies of the system, it is necessary to find the exact value of the effective mechanical stiffness (Km ) and unbiased mechanical resonance frequency of the system (f0 ). The unbiased resonance frequency of the in-phase vibration mode for DETF resonator can be estimated from: f0 =
1 2
Keff Meff
(7)
With the inclusion of electrical stiffness, the biased resonance frequency of the system can be calculated from:
f = f0
1+
Ke Km
(8)
Fig. 6 illustrates the simulation results for bidirectional control of fc when the coupling voltage is varied. The data was obtained for VBias = 110 V without stress on the resonators. The relatively large voltage needed here is due to the limit on the minimum gap distance (i.e., 2 m) in the fabrication technology and can be reduced significantly if devices with smaller gaps are fabricated. Simulations on the lumped model were carried in CoventorWare [21]. To illustrate the change in the location of the peaks, we set the damping coefficient in the simulation to a low value to have clear sharp peaks. However, such response is undesired for a bandpass filter and our fabricated devices do not exhibit this. As it can be seen in Fig. 6, changing the electrical spring stiffness of both resonators by the same value moves the resonance frequencies of
Fig. 7. The experimental setup for testing the coupled micro-resonators.
them together to the left or right depending on the value and sign of the coupling bias voltage (i.e., BW remains constant). Experiments were performed inside a vacuum chamber using a Rohde & Schwarz ZVB4 vector network analyzer (VNA) in a configuration as shown in Fig. 7. Fabricated devices were packaged in an 84 pin ceramic PGA package and placed inside a vacuum chamber at an operating pressure of about 1 Torr. A transimpedance amplifier with a gain of 100 k was used to amplify the current signal from the sense electrodes of the micro-resonators. Port 1 of the VNA was used to drive the resonators while the signal at the output of the amplifier was connected to Port 2. Experimental results for bi-directional tuning of fc through changing the coupling voltage (VCoupling ) are shown in Fig. 8 and are summarized in Table 2 (VBias = 110 V, VStress = 0). As can be seen, the filter exhibits low ripple (i.e., less than 1 dB) over its passband even though it is terminated to virtual ground at the input of the amplifier. This can be explained by noting the relatively low Q of the resonators (about 5000). On the other hand, due to the fabrication limitations, the nominal gap between the electrodes was about 2.5 m. These factors contributed to a motional resistance of about 550 k for each one of the DETFs, making proper termination challenging, especially since the signal loss through the filter is a function of various damping mechanisms. For a practical implementation of micromachined filters using electrostatic resonators, one needs to minimize the gap between the electrodes and try to
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Fig. 8. Experimental data for bidirectional tuning of fc by applying a DC coupling voltage on the coupling electrode.
Table 2 Effect of the coupling voltage on the experimental and analytical values of resonant frequencies. VCoupling (V)
VStress (V)
Change in BW (Hz)
Analytical
Experimental
302.987 303.281 303.563 303.830 304.090 304.335 304.568
302.993 303.289 303.569 303.833 304.081 304.329 304.561
0 60 85 110 135 160 185
4. Tuning filter bandwidth Changing the electrical stiffness gives a reasonable range of tunability in BW and fc of the coupled structures. However, both of these parameters change at the same time while the amplitude of the transmitted signal is also affected. On the other hand, small changes in axial stress on the resonators will not considerably affect the transmission magnitude [22]. Applying a tensile stress onto the structure increases the spring constant of the system. The applied axial force (F) modifies the spring stiffness (Kb ) of a Clamped–Clamped (CC) beam according to [23]: 4EI 3 L − 4 tanh(L/4)
(9)
where E is the Young’s modulus of elasticity, I is geometric moment of inertia, L is the length of the beam, and 2 = F/EI. Using Taylor’s series for small L, Eq. (9) can be simplified to:
L2 2 1+ 40
Analytical
Experimental
0 20 40 67 101 142 190
0 20 38 62 90 118 148
The resonance frequency of a CC beam with a mass attached to its centre is found from:
keep the environmental conditions as constant as possible so that proper termination of the filter is possible. For the unbiased system, it is not possible to measure the resonance frequencies of the in-phase vibration mode. In this mode, the anchor of the DETF also shows some compliance that is hard to be modelled. Moreover, fabrication tolerances also change the resonance frequency of the system. Therefore, experimental resonance frequencies of the biased system were substituted in (8) to estimate the unbiased resonance frequency. After calculating parameters f0 and Km , it is possible to analytically predict the resonance frequencies for the biased system. The small deviation in calculation of resonance frequency is related to the effect of coupling spring (KC ) that slightly changes the BW.
192EI Kb = L3
Table 3 Effect of VStress on bandwidth.
fc (kHz)
−15 −10 −5 0 +5 +10 +15
Kb =
Fig. 9. Tuning bandwidht through changing VStress .
(10)
1 f0 = 2
Kb Mp + 0.38 M
(11)
where Mp and M are the mass of the attached mass and beam respectively. The resonance frequency after applying the stress will be:
f = f0
1+
0.3L2 F EhW 3
(12)
where f0 is the resonance frequency of system at rest. Electrostatic actuators were employed using two symmetrical electrodes at the beam-ends (see Fig. 3) in order to apply an axial stress on the resonator beam. Using Taylor’s series again and substituting electrostatic force in (12) gives an analytical formula for change in resonance frequency as a function of applied voltage for generating the electrostatic force. Considering that each resonator in DETF sees half of the force, the change in bandwidth is found from:
BW = f0
0.075L2 EhW 3
ε0 A 2g 2
V2
(13)
where ε0 is the permittivity of air. Moreover, A, L, h and W are the area, length, height and width of electrodes, g is the gap and V is the difference in applied voltages between electrodes. Eq. (13) shows that the change in resonance frequency, and therefore filter BW, is linearly proportional to the applied axial force or quadratically to the applied voltage on the stress induction electrodes [22]. Fig. 9 illustrates the experimental results with VBias = 110 V, VCoupling = 0 for the change in BW as a function of VStress . For the particular device shown in Fig. 3, BW could be modified from 250 Hz to 400 Hz with less than 1 dB ripple (rPB ) in the passband of the filter. As expected, the signal attenuation is not significantly affected as the BW is changed. Table 3 compares the analytic and experimental results for BW adjustment using this technique. These results show that in low stress values, experimental results are very close to the analytical results from (13). One of
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it also makes it possible to properly terminate the filter, and hence, offering better passband characteristics. Acknowledgments The authors would like to acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), Western Economic Diversification (WD), and Simon Fraser University. Fabrication services were provided through CMC Microsystems. References
Fig. 10. Optimization of coupling between the microresonators for maximum BW with fixed VStress = 30 V.
the reasons for unmatched data for high voltages (high stress) could be ignoring the nonlinear mechanical effects. Another important factor that should be studied is sensitivity of this method for changing BW. In [9] we introduced a signal processing method for analysis of perturbations in arrays of coupled resonators. It can be shown that an optimum value exists for the coupling strength between resonators, which results in the highest sensitivity in changing the system poles to the inserted perturbation. Fig. 10 shows that by setting the bias voltages of the resonators independently, it is possible to modify the strength of coupling between them and hence getting the maximum change in BW for a given amount of stress. 5. Conclusions In this paper, a micromechanical filter design was proposed with independent tuning of its centre frequency and bandwidth. Since the principal idea of this filter is based on an array of weakly coupled micro-resonators, changing the coupling value modifies the centre frequency of the filter without affecting the separation between the resonance frequencies of the resonators. On the other hand, bandwidth of the system was changed by applying an axial stress on one of the resonators, which then modified its mechanical spring constant. One of the advantages of this method is the nearly constant transmission losses as the bandwidth is modified. We also presented the experimental results for an array of two electrically coupled DETF resonators, confirming that both filter parameters could be adjusted as the analytic models predicted. The filter performance can be improved significantly if it is fabricated in a process technology that is capable of reducing the electrostatic gaps. This not only reduces the losses through the resonators, but
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