Motional Capacitance Research Articles

A method of measuring equivalent parameters of quartz crystal resonators

Quartz crystal resonators (QCRs) can be described by four parameters in the BVD-equivalent circuit: static capacitance (C 0), motional capacitance (C 1), motional inductance (L 1), and motional resistance (R 1). In this study, we propose a set of formulae through theoretical deduction to evaluate these four parameters using the impedances of QCRs at different frequencies. Using this method, engineers can evaluate the four parameters without theoretical errors in only one frequency sweep. The method has been verified by LTspice simulation. In the simulation, the average error was 0.35%, which demonstrated its correctness. Meanwhile, in practical measurements, compared to the parameters reported by 250B, the parameters calculated using this method demonstrated better fitting ability. Moreover, the calculation process was simple. Therefore, the measuring procedures and apparatus can be simplified using this method.

Analysis and Design Considerations for Achieving the Fundamental Limits of Phase Noise in mmWave Oscillators With On-Chip MEMS Resonator

Very small electromechanical coupling coefficient in micro-electromechanical systems (MEMS) or acoustic resonators is quite a concern for oscillator performance, specially at mmWave frequencies. This small coefficient is the manifestation of the small ratio of motional capacitance to static capacitance in the resonators. This work provides a general solution to overcome the problem of relatively high static capacitance at mmWave frequencies and presents analysis and design techniques for achieving extremely low phase noise and a very high figure-of-merit (FoM) in an on-chip MEMS resonator based mmWave oscillator. The proposed analysis and techniques are validated with design and simulation of a 30 GHz oscillator with MEMS resonator having quality factor of 10,000 in 14 nm GF technology. Post layout simulation results show that it achieves a phase noise of −132 dBc/Hz and FoM of 217 dBc/Hz at offset of 1 MHz.

Piezoelectric Resonators Excited by Lateral Electric Fields Based on a LiTaO3 Single Crystal

In the present study, piezoelectric resonators under lateral field excitation (LFE) based on a LiTaO3 single crystal are modeled and analyzed. An electrically forced vibration study is employed to acquire the motional capacitance curve and vibration mode shapes. A finite element approach is utilized to investigate the influences of some basic parameters, such as the electrode/plate mass ratio, electrode gap, and electrode width on resonance characteristics. In addition, the design criteria for the gap and width of the electrode of the LiTaO3 LFE resonators are obtained by analyzing the effects of those parameters on vibration strain distributions. The obtained results are essential for designing LFE piezoelectric resonators by using a LiTaO3 single crystal.

Low-phase-noise self-sustaining amplifier IC with parallel capacitance cancellation for low-Q piezoelectric resonator

In this paper, a low-phase-noise self-sustaining amplifier IC with parallel capacitance cancellation for low-Q piezoelectric resonator is presented. The target of the proposed low-phase-noise self-sustaining amplifier IC is a mass-sensitive oscillator based on an AlN piezoelectric nanoresonator in liquid media. The behavioral model of the AlN piezoelectric nanoresonator is modeled as a damped second-order mass-spring-damper system with Verilog-A. The Verilog-A model enables the co-simulation of the oscillator nanosystem including the electronic sustaining amplifier circuit and the piezoelectric nanoresonator. The sustaining amplifier consists of two parts: transimpedance amplifier and shunt-capacitance cancelling amplifier. The shunt-capacitance cancellation and self-sustaining oscillation are critical in low-Q resonators, such as a mass sensor in liquid media. The shunt-capacitance-cancelling amplifier, which is a parasitic capacitance-canceller, supplies an inverted driving voltage to the nanoresonator to remove the wrong oscillation condition arising from the capacitance parallel to the nanoresonator. Near the resonant frequency, the motional inductance and motional capacitance of the nanoresonator are mutually cancelled, and the motional resistance are converted to the output voltage of the transimpedance amplifier. To remove the unwanted high-frequency poles, the amplifiers are designed using an inverter-based high-speed architecture with a 3 GHz gain-bandwidth product. In this oscillator system, when the target mass is attached to the nanoresonator, the inductance is increased; thus, the oscillation frequency is decreased. The operation of the full nanosystem is modeled and simulated using the Verilog-A behavioral model. The nominal output frequency is 5 MHz. The power consumption is 5 mA with 1.8 V supply voltage.

A novel langasite crystal microbalance instrumentation for UV sensing application

In this paper, we present a novel low-power UV sensor instrumentation based on ZnO thin-film and langasite crystal microbalance (LCM) composite resonator. The design of this sensor utilizes the exceptional transient response characteristics of thickness shear mode langasite crystal and UV sensitivity of annealed ZnO thin film. Our comparative transient analysis of langasite and quartz crystals shows that langasite crystal has high stability and fast relaxation times, and requires a fraction of excitation power in comparison to that of quartz. Upon investigating the equivalent circuit components, we discovered that high value of motional capacitance of LCM resulted in these contrasting transient responses in LCM and QCM. This high motional capacitance can be further attributed to the high electromechanical coupling coefficient (K2) of piezoelectric langasite crystal. Motivated by these observed characteristics, we proposed a novel sensor instrumentation that is simple yet effective in simultaneous measurements of the frequency, dissipation factor, and the amplitude of oscillation of a LCM. Moreover, our measurements allow for direct monitoring of any change in piezoelectricity of ZnO thin film. Our new measurement approach uses a one-shot pulse instead of conventional sinusoidal waveform to drive the crystal, which allows for simple instrumentation and requires very low input power. The transient response of the crystal shows that the crystal oscillates at its fundamental frequency and the amplitude of oscillation decays exponentially. From the recorded decay curve, the frequency of the freely oscillating crystal, and other important crystal parameters were calculated. We have utilized this mechanism to demonstrate highly sensitive ZnO thin film and LCM based UV sensor. Highest sensitivity of 35.8ppm per μW/cm2 was observed for the LCM coated with 400nm ZnO film. Additionally, the influence of UV on the piezoelectric output of ZnO thin film was also characterized by monitoring the DC offset of the output oscillations. Our findings in this work opens scope for developing low-power acoustic sensors with significant implications towards self-powering sensors.

Quantum electromechanics on silicon nitride nanomembranes.

Radiation pressure has recently been used to effectively couple the quantum motion of mechanical elements to the fields of optical or microwave light. Integration of all three degrees of freedom—mechanical, optical and microwave—would enable a quantum interconnect between microwave and optical quantum systems. We present a platform based on silicon nitride nanomembranes for integrating superconducting microwave circuits with planar acoustic and optical devices such as phononic and photonic crystals. Using planar capacitors with vacuum gaps of 60 nm and spiral inductor coils of micron pitch we realize microwave resonant circuits with large electromechanical coupling to planar acoustic structures of nanoscale dimensions and femtoFarad motional capacitance. Using this enhanced coupling, we demonstrate microwave backaction cooling of the 4.48 MHz mechanical resonance of a nanobeam to an occupancy as low as 0.32. These results indicate the viability of silicon nitride nanomembranes as an all-in-one substrate for quantum electro-opto-mechanical experiments.

A semi-analytical modeling approach for laterally-vibrating thin-film piezoelectric-on-silicon micromechanical resonators

In this paper, we propose an analytical model that more accurately captures the electro-mechanical behavior of laterally-vibrating extensional-mode thin-film piezoelectric-on-silicon resonators. The proposed modeling approach considers the actual vibration mode shape of the resonator instead of relying on simplified modal functions such as assuming that the displacements occur only within the plane of fabrication and follow a sine function in the vibration plane. Out-of-plane and planar deviations in the mode shape from an ideal sine function arise from differences in the acoustic velocities between the piezoelectric film and underlying silicon. The proposed model allows more accurate prediction of the key lumped parameters in both the mechanical (e.g. equivalent point force associated with the reverse piezoelectric effect, effective stiffness) and electrical domain (i.e. motional capacitance) as a function of electrode coverage. Compared to an existing 1D approximation model proposed in a seminal work, the lumped parameters derived from the proposed model show notably closer agreement with 3D coupled-domain finite element (FE) simulations and also the measured results from fabricated devices. The semi-analytical nature of our modeling approach relies on the details of the mode shape provided. As such, our model has the benefit of allowing increasing levels of precision as desired depending on the amount of detail that is included in the FE modal analysis.

Resonances and energy trapping in AT-cut quartz resonators operating with fast shear modes driven by lateral electric fields produced by surface electrodes

We analyze coupled thickness-shear and extensional vibrations of a piezoelectric resonator of AT-cut quartz. Different from most of the AT-cut quartz resonators studied in the literature which are based on the slow shear mode excited by a thickness electric field, the resonator in this paper operates with the fast shear mode driven by a lateral electric field produced by a pair of electrodes on the top surface of the resonator. Mindlin’s first-order theory of piezoelectric plates is used. Dispersion relations of the relevant waves in unelectroded and electroded plates are presented and compared. The motional capacitance, resonant frequencies and mode shapes near resonances are obtained from an electrically forced vibration analysis. Trapped modes without vibration near the resonator edges are identified. The effects of various structural parameters on energy trapping are examined and the mechanisms are discussed. The results can provide important bases for the parameters design of new resonators operating with the fast shear mode with new excitation schemes.

Analysis of change in motional capacitance of electrical equivalent circuit of quartz-crystal tuning-fork tactile sensor induced by viscoelastic materials in contact with its base

We investigated experimentally and theoretically the change in the reciprocal of the motional capacitance Δ(1/Ca) of a quartz-crystal tuning-fork tactile sensor before and after its base comes into contact with neoprene rubbers. We derived the analytical formula for the motional capacitance of the electrical equivalent circuit of the sensor at resonance based on the combination of the L-shaped bar model and viscoelastic foundation model, and the law energy of conservation. We found from both contact experiments and theoretical considerations on the analytical formula that Δ(1/Ca) is intrinsically induced by both the dynamic Young’s modulus and viscosity of neoprene rubbers at 32.5 kHz, which is the actual resonant frequency of the quartz tactile sensor. In this paper, we showed experimentally and theoretically for the first time that the motional capacitance of the tuning fork tactile sensor is affected by the dynamic viscosity of materials in contact with the sensor’s base, except their dynamic Young’s modulus.

1P3-18 Analysis of Change in Motional Capacitance of Quartz-Crystal Tuning-Fork Tactile Sensor Induced by Viscoelastic Materials in Contact with Its Base(Poster Session)

1P3-18 Analysis of Change in Motional Capacitance of Quartz-Crystal Tuning-Fork Tactile Sensor Induced by Viscoelastic Materials in Contact with Its Base(Poster Session)

Series resistance method to obtain equivalent circuit of piezoelectric resonator

The series resistance method (SRM) is introduced to obtain an equivalent circuit of a piezoelectric micro resonator. This method uses two series resistances respectively connected to driving electrodes of the resonator, and then detects the voltage amplitudes at the terminals of the series resistances and phase difference between them. Compared with the results of the impedance analyser, the deviations of resonance frequency, motional capacitance (C 1 ), motional inductance (L 1 ) of the SRM are 0.213%, 7.48%, and 7.62%, respectively. The SRM is easy to be realised in the laboratory. To a certain extent, it can replace the impedance analyser to attain the equivalent parameters of the piezoelectric micro resonator.

High-frequency vibrations of piezoelectric plates driven by lateral electric fields

We study coupled face-shear and thickness-twist motions of piezoelectric plates of monoclinic crystals driven by lateral electric fields. The first-order theory of piezoelectric plates is used. Pure thickness modes and propagating waves in unbounded plates as well as vibrations of finite plates are studied. Both free vibrations and electrically forced vibrations are considered. Basic vibration characteristics including resonant frequencies, dispersion relations, frequency spectra and motional capacitance are obtained. Numerical results are presented for AT-cut quartz plates. The results are expected to be useful for the understanding and design of resonant piezoelectric devices using lateral field excitation.

Direct parameter extraction in feedthrough-embedded capacitive MEMS resonators

This paper presents a method for fast and accurate determination of parameters relevant to the characterization of capacitive MEMS resonators like quality factor ( Q), resonant frequency ( f n ), and equivalent circuit parameters such as the motional capacitance ( C m ). In the presence of a parasitic feedthrough capacitor ( C F ) appearing across the input and output ports, the transmission characteristic is marked by two resonances: series ( S) and parallel ( P). Close approximations of these circuit parameters are obtained without having to first de-embed the resonator motional current typically buried in feedthrough by using the series and parallel resonances. While previous methods with the same objective are well known, we show that these are limited to the condition where C F ≪ C m Q. In contrast, this work focuses on moderate capacitive feedthrough levels where C F > C m Q, which are more common in MEMS resonators. The method is applied to data obtained from the measured electrical transmission of fabricated SOI MEMS resonators. Parameter values deduced via direct extraction are then compared against those obtained by a full extraction procedure where de-embedding is first performed and followed by a Lorentzian fit to the data based on the classical transfer function associated with a generic LRC series resonant circuit.

Static and Motional Feedthrough Capacitance of Flexural Microresonator

The present paper evaluates the static and motional feedthrough capacitance of a silicon carbide-based flexural-mode microelectromechanical system resonator. The static feedthrough capacitance was measured by a network analyzer under atmospheric pressure. The motional feedthrough was obtained by introducing various values into the modeling circuit in order to fit the Bode plots measured under reduced pressure. The static feedthrough capacitance was 0.02 pF, whereas the motional feedthrough capacitance of an identical device was about 0.2 pF, which is one order of magnitude larger than the static feedthrough capacitance.

3Pa3-4 Elucidation of Change in Motional Capacitance of Quartz-Crystal Tuning-Fork Tactile Sensor Induced by Viscoelastic materials in Contact with Its Base(Poster Session)

3Pa3-4 Elucidation of Change in Motional Capacitance of Quartz-Crystal Tuning-Fork Tactile Sensor Induced by Viscoelastic materials in Contact with Its Base(Poster Session)

Analysis of Change in Motional Capacitance of the Electrical Equivalent Circuit of Quartz-Crystal Tuning-Fork Tactile Sensor at Resonance by Combining Analytical Models for Elastic Foundation and L-Shaped Bar with Torsion Spring

We derived the analytical formula of the motional capacitance Ca of a quartz-crystal tuning-fork tactile sensor by use of the conservation law of energy that the electrostatic energy in Ca is equal to the sum of the strain energy of its arm and the elastic energy of the material in contact with its base. In this analysis, the strain energy of the arm and the elastic energy of the material in contact with its base were obtained by applying the torsion spring model to the joint of the arm and the base, which were depicted as an L-shaped bar, and the elastic foundation to the materials in contact with its base, and the electrostatic energy of the arm was derived by applying the bimorph flexural model to the arm treated as a piezoelectric material. As a result of calculation, we found that the calculated value of the change in reciprocal motional capacitance Δ(1/Ca) of the quartz-crystal tuning-fork tactile sensor by our model is closer to the measured one for plastics than that given by the theoretical formula of motional capacitance of only one arm and Δ(1/Ca) is affected by the Young's moduli of materials in contact with the sensor's base.

Energy trapping in high-frequency vibrations of piezoelectric plates with partial mass layers under lateral electric field excitation

We study coupled face-shear (FS) and thickness-twist (TT) motions of a piezoelectric plate of monoclinic crystals with mass layers on the central parts of the plate surfaces. The plate is driven by a lateral electric field. Mindlin’s first-order theory of piezoelectric plates is used. An analytical solution is obtained. Numerical results are presented for an AT-cut quartz plate, including the motional capacitance of the plate as a resonator and the vibration modes trapped under the mass layers in the central portion of the plate. The relationship between the dimension of the mass layers and the number of trapped modes is examined.

Proposal of Novel Method to Measure Young's Modulus of Materials Using Change in Motional Capacitance of the Electrical Equivalent Circuit of Quartz-Crystal Tuning-Fork Tactile Sensor at Resonance

We propose a novel method to measure the Young's modulus of materials using the change in motional capacitance of the electrical equivalent circuit of a quartz-crystal tuning-fork tactile sensor at resonance before and after the sensor's base coming into contact with materials with different Young's moduli, such as silicon rubbers and plastics. How to measure the Young's modulus of materials using the change in motional capacitance is investigated experimentally and theoretically. By our calculation of motional capacitance and from contact experiments on silicon rubbers and plastics, we found that there was a possibility to measure the Young's modulus of materials using the relationship between the change in motional capacitance and their Young's modulus.

On the inaccuracy of using mindlin's first- order plate theory for calculating the motional capacitance of a thickness-shear resonator

We show that when using Mindlin's first-order theory of piezoelectric plates to analyze a thickness-shear mode resonator, while the frequency can be predicted accurately, the charge, the current, and hence the motional capacitance have significant inaccuracy.

Development of SMD 32.768 kHz tuning fork-type crystals

Abstract Taguchi' design of experiment method and statistical F-test were used to optimize design of tuning forks for use in Qualcomm® mobile station modem (MSM)-3000TM series central processing unit chips of code division multiple access, personal communication system, and global system for mobile communication systems. Design parameters for motional capacitance C1 was further optimized after much trial and error so that as-fabricated resonance frequency values of a tuning fork blank remain almost the same prior to final frequency adjustment plating of the mounted crystals, regardless of packaging pressure change from ambient to designated 6 × 10– 5 Torr. Finite element method modeling was used in conjunction with Taguchi' design of experiment method and, after much trial and error, this enabled selection of essential design parameters to effectively manufacture tuning fork test samples using photolithography with subsequent photoresist spray coating. The tuning fork test samples thus fabricated were characterized and analyzed using Taguchi' method and statistical F-test to evaluate sensitivity of motional capacitance C1 and as-fabricated resonance frequency for various design parameters at ambient to designated packaging pressure of 6 × 10– 5 Torr. Motional capacitance C1 is most sensitive to tine width and tine surface electrode width. Unlike as-fabricated resonance frequency values, motional capacitance values do change with vacuum level of the package. However, tine width asymmetry, electrode thickness and tine tip electrode thickness could be freely adjusted for crystal impedance level without affecting resonance frequency and motional capacitance value. Other design parameters could be adjusted to best suit the manufacturing process.