Analog circuit of weakly coupled fractional-order MEMS resonator

Abstract

This invention relates to an analog circuit for a weakly coupled fractional-order MEMS resonator, belonging to the field of resonator circuit design. The circuit includes four proportional-integral (PI) operational circuits, four inverting proportional circuits, fourteen multipliers, two DC power supplies, and two AC power supplies. Each PI operational circuit comprises a fractional-order unit circuit, a first operational amplifier, and one resistor. The fractional-order unit circuit is composed of three sets of parallel capacitor-resistor structures connected in series. This invention improves the accuracy of the analog circuit for the weakly coupled fractional-order MEMS resonator, providing strong technical support for subsequent hardware-level engineering development.

Description

Technical Field

[0001] This invention belongs to the field of resonator circuit design and relates to an analog circuit for a weakly coupled fractional-order MEMS resonator. Background Technology

[0002] MEMS resonators, due to their low noise, low power consumption, wide dynamic range, and ability to be integrated into IC circuits, have been widely used in sensors, navigation systems, mobile phones, smart vehicles, and orbital satellites. However, single-crystal silicon crystals are highly sensitive to common-mode interference, vibration amplitude, and environmental factors such as temperature and pressure. System modeling plays a fundamental and crucial role in the circuit implementation of weakly coupled MEMS resonators. Numerous scholars have studied integer-order modeling and nonlinear analysis of individual MEMS resonators. As representatives of this research, Mesrom et al. established a mechanical model of a single MEMS resonator and estimated the impact of thermal noise by constructing a measurement circuit; however, these works do not meet current performance requirements such as high sensitivity and noise immunity. Scattered records in existing literature have verified the application of fractional-order calculations in MEMS resonators. Fitt et al. solved the control equations of MEMS resonators using fractional-order differential equations. Aghababa studied the existence of chaos in MEMS resonators and proposed a fractional-order finite-time controller to suppress their inherent oscillations. These valuable attempts provide some insights for this research. However, they did not establish a structure between coupling strength and oscillator dissipation levels, nor did they conduct in-depth research on circuit design.

[0003] Constructing reasonable electronic circuits at the hardware level is a time-efficient and controllable method. The constructed circuits can comprehensively scan the parameter space and detect relevant values ​​of the dynamic performance of electromechanical systems. For this reason, it has received great attention in experimental verification and rapid application development. Sabarathinam and Thamilmaran built the Duffing series of MEMS resonators using analog circuits and proved their inherent chaotic oscillations. Zhang et al. constructed analog circuits and proposed four neural network inversion control methods for Duffing-type MEMS resonators. Luo et al. designed the electronic circuits for a coupled fractional-order self-sustaining electromechanical seismograph system and provided a neural adaptive optimal fixed-time synchronization scheme. However, such analog circuits are susceptible to electronic instrument errors and external interference, significantly affecting the real-time performance of data transmission. Furthermore, large signal amplitudes can lead to saturation distortion in multipliers and operational amplifiers, lacking versatility and flexibility.

[0004] Therefore, a new analog circuit is urgently needed to solve the above problems. Summary of the Invention

[0005] In view of this, the purpose of the present invention is to provide an analog circuit for a weakly coupled fractional-order MEMS resonator, thereby improving the accuracy of the analog circuit.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] An analog circuit for a weakly coupled fractional-order MEMS resonator includes four proportional-integral operational circuits based on TL074CN op-amps, four inverting proportional circuits based on TL074CN op-amps, fourteen AD633JN multipliers (A1 to A14), two DC power supplies, and two AC power supplies.

[0008] The proportional-integral operation circuit and the inverting proportional circuit are connected in series through resistors (R4, R7 and R10, or R19, R22 and R24); wherein the output terminals of the first to fourth proportional-integral operation circuits are respectively connected to voltage signals -x2, -x1, -x4 or -x3, and the output terminals of the first to fourth inverting proportional circuits are respectively connected to voltage signals x2, x1, x4 or x3.

[0009] The x-input terminal of multiplier A2 is connected to the output terminal of multiplier A1, and the y-input terminals of multipliers A5 and A7. The y-input terminal of multiplier A2 is connected to the voltage signal -x1. The output terminal of multiplier A2 is connected to the input terminal of the first proportional-integral (PI) circuit through resistor R2. Both input terminals of multiplier A1 are connected to the voltage signal x1 and are also connected to the input terminal of the first PI circuit through resistor R13. The x-input terminal of multiplier A3 is connected to the x-input terminals of multipliers A4 and A5, the y-input terminal of multiplier A6, and the AC power supply V1. The y-input terminal of multiplier A3 is connected to the DC power supply V5. The output terminal of multiplier A3 is connected to the AC power supply V5 through resistor R15. The first proportional-integral (PI) circuit input is connected to the input terminal of the first PI circuit; the y-input terminal of multiplier A4 is connected to the voltage signal x1, and the output terminal of multiplier A4 is connected to the input terminal of the first PI circuit through resistor R16; the output terminal of multiplier A5 is connected to the input terminal of the first PI circuit through resistor R17; the x-input terminal of multiplier A6 is connected to the voltage signal x1, and the output terminal of multiplier A6 is connected to the x-input terminal of multiplier A7; the output terminal of multiplier A7 is connected to the input terminal of the first PI circuit through resistor R18; the input terminals of the first PI circuit are connected to the voltage signals -x2 and x1 respectively through resistors R1 and R9.

[0010] The x-input terminal of multiplier A9 is connected to the y-input terminals of multipliers A12 and A14, and the output terminal of multiplier A8. The x-input terminal and y-input terminal of multiplier A9 are connected to the voltage signal -x3. The output terminal of multiplier A9 is connected to the input terminal of the third proportional-integral (PI) circuit through resistor R30. Both input terminals of multiplier A8 are connected to the voltage signal x3. The x-input terminal of multiplier A10 is connected to the x-input terminals of multipliers A11 and A12, and the y-input terminal of multiplier A13. The y-input terminal of multiplier A10 is connected to the DC power supply V5. The output terminal of multiplier A10 is connected to the third PI circuit through resistor R33. The circuit input terminals are as follows: the y-input terminal of multiplier A11 is connected to the voltage signal x3, and the output terminal of multiplier A11 is connected to the input terminal of the third proportional-integral (PI) circuit through resistor R34; the output terminal of multiplier A12 is connected to the input terminal of the third PI circuit through resistor R35; the x-input terminal of multiplier A14 is connected to the output terminal of multiplier A13, and the output terminal of multiplier A14 is connected to the input terminal of the third PI circuit through resistor R36; the input terminals of the third PI circuit are connected to the voltage signals -x4, x3, -x1, and x3 respectively through resistors R27, R29, R31, and R32.

[0011] Preferably, the proportional-integral operational circuit includes a fractional-order unit circuit, an operational amplifier I (U1A, U3A, U5A, or U7A), and a resistor (R3, R8, R28, or R23); one end of the fractional-order unit circuit is connected to the negative input terminal of the operational amplifier I, and the other end is connected to the output terminal of the operational amplifier I; one end of the resistor (R3, R8, R28, or R23) is connected to the positive input terminal of the operational amplifier I, and the other end is grounded.

[0012] The fractional-order unit circuit is composed of three sets of parallel capacitors (C1~C3, C4~C5, C7~C9 or C10~C12) and resistors (R37~R39, R40~R42, R43~R45 or R46~R48) connected in series.

[0013] Preferably, the inverting amplifier circuit includes an operational amplifier II (U2A, U4A, U6A, or U8A) and two resistors (R5-R6, R11-R12, R20-R21, or R25-R26); one end of the resistor (R5, R11, R20, or R25) is connected to the positive input terminal of the operational amplifier II, and the other end is grounded; one end of the resistor (R6, R12, R21, or R26) is connected to the negative input terminal of the operational amplifier II, and the other end is connected to the output terminal of the operational amplifier II.

[0014] Preferably, the voltage signals (x1, x2, x3, x4) are mappings to the state variables (x1, x2, x3, x4) of a weakly coupled MEMS resonator; using Kirchhoff's circuit laws and constitutive relations, the relevant circuit equations are written as follows:

[0015]

[0016]

[0017]

[0018]

[0019] Where F(s) represents the transfer function of the fractional-order unit circuit.

[0020] Furthermore, the expression for the transfer function F(s) of the fractional-order unit circuit is:

[0021]

[0022] Among them, C o For specific capacitance, C a C b and C c For the selected capacitor, R a R b and R c Let be the selected resistor, and s be the complex frequency.

[0023] The beneficial effects of this invention are as follows: This invention improves the accuracy of the simulation circuit of the weakly coupled fractional-order MEMS resonator, and provides strong technical support for subsequent hardware-level engineering development.

[0024] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0025] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0026] Figure 1 The schematic diagram (a) of a weakly coupled MEMS resonator and its equivalent spring-mass-damping model (b) are shown.

[0027] Figure 2 For α = 0.993, V AC =0.4 and V ACFractional-order analog circuit diagram of a weakly coupled MEMS resonator with a capacitance of 0.4.

[0028] Figure 3 The phase diagram is for a fractional-order analog circuit of a weakly coupled MEMS resonator. Detailed Implementation

[0029] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of the present invention. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0030] The accompanying drawings are for illustrative purposes only and are schematic diagrams, not actual pictures. They should not be construed as limiting the invention. To better illustrate the embodiments of the invention, some parts in the drawings may be omitted, enlarged, or reduced, and do not represent the actual product dimensions. It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings.

[0031] In the accompanying drawings of the embodiments of the present invention, the same or similar reference numerals correspond to the same or similar components. In the description of the present invention, it should be understood that if terms such as "upper," "lower," "left," "right," "front," and "rear" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, they are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, the terms used to describe positional relationships in the drawings are only for illustrative purposes and should not be construed as limiting the present invention. For those skilled in the art, the specific meaning of the above terms can be understood according to the specific circumstances.

[0032] Please see Figures 1-3 :

[0033] 1. System Modeling:

[0034] Figure 1 The modeling process of a weakly coupled fractional-order MEMS resonator is shown. The principle of the weakly coupled MEMS resonator fabricated using the standard silicon-on-insulator (SiI) process is as follows: Figure 1 As shown in (a). The corresponding process includes multiple steps such as photoresist deposition and etching, deep reactive ion etching, metal electrode etching, and carrier wafer bonding. From Figure 1As shown in (a), two comb-driven and sensing MEMS resonators are coupled by bridge coupling beams with low stiffness on the top and bottom sides. Two identical inertial mass blocks are placed on the left and right sides of MEMS resonators 1 and 2, respectively, and suspended from a folded suspension beam. Driving the comb capacitors allows the MEMS resonators to generate a large linear displacement, which is sensed by the sensing comb capacitors, resulting in a large output signal. The AC drive port provides the MEMS resonators with a heterogeneous electrostatic driving force. When acceleration acts on the inertial mass, the weakly coupled MEMS resonator generates a differential negative stiffness perturbation. Model localization causes a drastic change in the amplitude ratio of the weakly coupled MEMS resonator. Sensing acceleration by measuring the displacement of the amplitude ratio provides good sensitivity.

[0035] The mass-spring-damper model of a weakly coupled MEMS resonator is as follows: Figure 1 As shown in (b), Westerlund and Ekstam used experimental results to reveal the inherent fractional-order characteristics of different capacitor dielectrics. Based on this, the total kinetic energy of a weakly coupled MEMS resonator is defined as...

[0036]

[0037] Where m1 and m2 represent the total mass of the first and second MEMS resonators, respectively, z1 and z2 represent the bending displacement, and α and This represents the fractional coefficient and time.

[0038] The potential energy of a weakly coupled fractional-order MEMS resonator is

[0039]

[0040] Where, k′ c The coefficient of friction is denoted by k′1(z1) and k′2(z2) represents the coupling stiffness between the two MEMS resonators, and k′1(z1) and k′2(z2) represent the nonlinear stiffness of each resonator.

[0041] The nonlinear stiffness containing second-order terms is

[0042]

[0043] Where, k′ i1 and k′ i2 These represent the second-order correction terms for the linear spring constant and the linear stiffness, respectively.

[0044] The sensitivity of weakly coupled MEMS resonators is closely related to the perturbation of electrostatic negative stiffness. When the inertial mass begins to accelerate, the electrostatic negative stiffness is expressed as...

[0045]

[0046] Among them, ε, S A , ΔV, g0, m p a and k p Let k represent the dielectric constant, the cross-sectional area of ​​the inductive parallel plate capacitor, the potential difference between the inductive mass and the weakly coupled MEMS resonator, the initial gap of the inductive parallel plate capacitor, the mass of the inductive mass block, and the acceleration and stiffness of the inductive mass suspension, respectively. Δ =-εS A ΔV 2 / g0.

[0047] Electrostatic driving force is written as

[0048]

[0049] in, V d =V DC The upper and lower electrode voltages applied to each MEMS resonator. C0, d0, z i and These represent capacitance, initial gap width, bending displacement, and frequency, respectively. DC and V AC This indicates the bias voltage and AC voltage.

[0050] The Taylor expansion of electrostatic driving force is:

[0051]

[0052] Here, hot represents a higher-order term.

[0053] Lagrange force L and generalized force Q i for

[0054]

[0055] Among them, c i Indicates damping.

[0056] Thus, the Lagrange equations of motion for the weakly coupled MEMS resonator are obtained.

[0057]

[0058] in, This represents the Caputo derivative with respect to the origin, under differential perturbation.

[0059] Introducing some dimensionless variables simplifies the mathematical model, such as: A i =2r i V AC / V DC , μ i =c i / m i ω0, x1 = z1 / d0, x3=z2 / d0 and Assume V AC Much smaller than V DC Furthermore, by introducing control input, equation (8) is rewritten as follows:

[0060]

[0061] Where u2 and u4 represent control inputs.

[0062] Note 1: Compared with a single resonator, a weakly coupled MEMS resonator can better improve system sensitivity. Furthermore, a reasonable selection of eigenstates or amplitude ratios closely related to coupling stiffness and electrostatic stiffness can further improve system sensitivity. Meanwhile, the differential perturbation scheme of the weakly coupled MEMS resonator is significantly better than the single-sided perturbation scheme in improving system sensitivity. Additionally, the fractional-order model (8) of the weakly coupled MEMS resonator has a longer memory dependency and higher design freedom. If α = 1, the fractional-order model will degenerate into an integer-order model.

[0063] 2. Circuit Design and Analysis:

[0064] To enable hardware verification of the aforementioned system and facilitate subsequent engineering development, a frequency approximation is used to characterize the fractional-order system within acceptable error limits, since fractional-order calculus operators cannot be directly calculated in time-domain simulations and experiments.

[0065] When the angular velocity range is

[10] -2 10 2 ]rad / s, with a maximum difference of 0.2dB in the frequency domain at 1 / s 0.993 The approximate transfer function is defined as:

[0066]

[0067] Where s is the complex frequency.

[0068] To realize the unit circuit of F(s) in the analog circuit, its transfer function, composed of resistors and capacitors, is defined as...

[0069]

[0070] Among them, C o For specific capacitance, C a C b and C c For the selected capacitor, Ra R b and R c The selected resistor.

[0071] Using equations (9) and (10), a fractional-order analog circuit for a weakly coupled MEMS resonator is established, such as... Figure 2 As shown, the circuit includes four proportional-integral (PI) operational amplifier (PI) circuits based on TL074CN op-amps, four inverting PI circuits based on TL074CN op-amps, fourteen AD633JN multipliers, two DC power supplies, and two AC power supplies. To construct the unit circuit for F(s), C is chosen... a = 9.93018 nF, C b =208.9nF, C c =199.7nF, R a = 983.949 MΩ, R b =62.0449kΩ and R c = 86.096Ω. Clearly, there are four fractional-order unit circuits, such as C1-R37-C2-R38-C3-R39, C4-R40-C5-R41-C6-R42, C7-R43-C8-R44-C9-R45, and C10-R46-C11-R47-C12-R48. The parameters of all electronic components are as follows: Figure 2 As shown.

[0072] The voltage signals (x1, x2, x3, x4) are mappings to the state variables (x1, x2, x3, x4) of a weakly coupled MEMS resonator. Using Kirchhoff's circuit laws and constitutive relations, the relevant circuit equations are written as follows:

[0073]

[0074] Figure 3 The phase diagram of the fractional-order analog circuit of the weakly coupled MEMS resonator was displayed using an oscilloscope, demonstrating the correctness of the constructed analog circuit.

[0075] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. An analog circuit for a weakly coupled fractional-order MEMS resonator, characterized in that, The circuit includes 4 proportional-integral operation circuits, 4 inverting proportional circuits, 14 multipliers (A1~A14), 2 DC power supplies and 2 AC power supplies. The proportional-integral operation circuit and the inverting proportional circuit are connected in series through resistors (R4, R7 and R10, or R19, R22 and R24); wherein the output terminals of the first to fourth proportional-integral operation circuits are respectively connected to voltage signals -x2, -x1, -x4 or -x3, and the output terminals of the first to fourth inverting proportional circuits are respectively connected to voltage signals x2, x1, x4 or x3; The x-input terminal of multiplier A2 is connected to the output terminal of multiplier A1, and the y-input terminals of multipliers A5 and A7. The y-input terminal of multiplier A2 is connected to the voltage signal -x1. The output terminal of multiplier A2 is connected to the input terminal of the first proportional-integral (PI) circuit through resistor R2. Both input terminals of multiplier A1 are connected to the voltage signal x1 and are also connected to the input terminal of the first PI circuit through resistor R13. The x-input terminal of multiplier A3 is connected to the x-input terminals of multipliers A4 and A5, the y-input terminal of multiplier A6, and the AC power supply V1. The y-input terminal of multiplier A3 is connected to the DC power supply V5. The output terminal of multiplier A3 is connected to the AC power supply V5 through resistor R15. The first proportional-integral (PI) circuit input is connected to the input terminal of the first PI circuit; the y-input terminal of multiplier A4 is connected to the voltage signal x1, and the output terminal of multiplier A4 is connected to the input terminal of the first PI circuit through resistor R16; the output terminal of multiplier A5 is connected to the input terminal of the first PI circuit through resistor R17; the x-input terminal of multiplier A6 is connected to the voltage signal x1, and the output terminal of multiplier A6 is connected to the x-input terminal of multiplier A7; the output terminal of multiplier A7 is connected to the input terminal of the first PI circuit through resistor R18; the input terminals of the first PI circuit are connected to the voltage signals -x2 and x1 respectively through resistors R1 and R9. The x-input terminal of multiplier A9 is connected to the y-input terminals of multipliers A12 and A14, and the output terminal of multiplier A8. The x-input terminal and y-input terminal of multiplier A9 are connected to the voltage signal -x3. The output terminal of multiplier A9 is connected to the input terminal of the third proportional-integral (PI) circuit through resistor R30. Both input terminals of multiplier A8 are connected to the voltage signal x3. The x-input terminal of multiplier A10 is connected to the x-input terminals of multipliers A11 and A12, and the y-input terminal of multiplier A13. The y-input terminal of multiplier A10 is connected to the DC power supply V5. The output terminal of multiplier A10 is connected to the third PI circuit through resistor R33. The circuit input terminals are as follows: the y-input terminal of multiplier A11 is connected to the voltage signal x3, and the output terminal of multiplier A11 is connected to the input terminal of the third proportional-integral (PI) circuit through resistor R34; the output terminal of multiplier A12 is connected to the input terminal of the third PI circuit through resistor R35; the x-input terminal of multiplier A14 is connected to the output terminal of multiplier A13, and the output terminal of multiplier A14 is connected to the input terminal of the third PI circuit through resistor R36; the input terminals of the third PI circuit are connected to the voltage signals -x4, x3, -x1, and x3 respectively through resistors R27, R29, R31, and R32.

2. The analog circuit of the weakly coupled fractional-order MEMS resonator according to claim 1, characterized in that, The proportional-integral operational circuit includes a fractional-order unit circuit, an operational amplifier I (U1A, U3A, U5A, or U7A), and a resistor (R3, R8, R28, or R23). One end of the fractional-order unit circuit is connected to the negative input terminal of the operational amplifier I, and the other end is connected to the output terminal of the operational amplifier I. One end of the resistor (R3, R8, R28, or R23) is connected to the positive input terminal of the operational amplifier I, and the other end is grounded.

3. The analog circuit of the weakly coupled fractional-order MEMS resonator according to claim 2, characterized in that, The fractional-order unit circuit is composed of three sets of parallel capacitors (C1~C3, C4~C5, C7~C9 or C10~C12) and resistors (R37~R39, R40~R42, R43~R45 or R46~R48) connected in series.

4. The analog circuit of the weakly coupled fractional-order MEMS resonator according to claim 1, characterized in that, The inverting amplifier circuit includes an operational amplifier II (U2A, U4A, U6A, or U8A) and two resistors (R5-R6, R11-R12, R20-R21, or R25-R26). One end of the resistor (R5, R11, R20, or R25) is connected to the positive input terminal of the operational amplifier II, and the other end is grounded. One end of the resistor (R6, R12, R21, or R26) is connected to the negative input terminal of the operational amplifier II, and the other end is connected to the output terminal of the operational amplifier II.

5. The analog circuit for a weakly coupled fractional-order MEMS resonator according to any one of claims 1 to 4, characterized in that, The voltage signals (x1, x2, x3, x4) are a mapping of the state variables (x1, x2, x3, x4) of a weakly coupled MEMS resonator; using Kirchhoff's circuit laws and constitutive relations, the relevant circuit equations are written as follows: Where F(s) represents the transfer function of the fractional-order unit circuit.

6. The analog circuit for a weakly coupled fractional-order MEMS resonator according to claim 5, characterized in that, The expression for the transfer function F(s) of a fractional-order unit circuit is: Among them, C o For specific capacitance, C a C b and C c For the selected capacitor, R a R b and R c Let be the selected resistor, and s be the complex frequency.

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