A MEMS macro model construction method based on modal superposition method and state space equation

Abstract

This invention discloses a method for constructing MEMS macro-models based on modal superposition and state-space equations. First, a finite element model is established and modal analysis is performed to obtain the system matrix. Second, the dominant mode is selected based on modal contributions. Then, multi-physics parameters such as capacitive sensing parameters, nonlinear stiffness based on implicit condensation, and damping coefficients are extracted in parallel. Next, these parameters are coupled in multiple physics and the model order is reduced. Finally, the model is converted into a state-space description and encapsulated using a hardware description language. The macro-model constructed by this invention can be seamlessly integrated with phase-locked loop (PLL) detection circuits. This invention achieves simulation acceleration of over 700 times while maintaining physical accuracy (with an error of less than 1.6% compared to 3D finite element analysis), efficiently evaluating nonlinear effects and temperature drift compensation characteristics, and significantly improving the design efficiency of complex microsystems.

Description

Technical Field

[0001] This invention belongs to the field of microelectromechanical systems (MEMS) design and simulation technology, and particularly relates to a method for constructing MEMS macro-models based on modal superposition and state-space equations. Background Technology

[0002] MEMS device design typically follows a decoupled approach, where device development and interface circuit design are performed independently. This decoupling presents significant challenges in system-level optimization because of the strong coupling effects between MEMS devices and their electronic interfaces, making prediction difficult using isolated design methods. This problem is particularly acute for MEMS resonant electric field microsensors (EFMs) due to the complex interactions between mechanical resonance, electrostatic induction, and electronic detection circuitry.

[0003] Traditional MEMS device modeling methods (including equivalent circuit models and lumped parameter models) face significant limitations. These methods require manually deriving complex behavioral equations to describe physical processes, becoming increasingly complex when dealing with nonlinear effects. Furthermore, the accuracy of these models heavily relies on the designer's experience and simplifying assumptions, often resulting in a lack of model universality and requiring extensive calibration. Summary of the Invention

[0004] The purpose of this invention is to provide a MEMS macro-model construction method based on modal superposition and state-space equations. By combining finite element analysis and order reduction algorithm for automated modeling, it can effectively handle nonlinear effects and seamlessly integrate with detection circuits for system-level co-simulation, thereby solving the technical problems mentioned in the background art.

[0005] To solve the above-mentioned technical problems, the specific technical solution of the present invention is as follows: A method for constructing MEMS macro-models based on modal superposition and state-space equations includes the following steps: Step 1: Finite element modeling: Construct a finite element model of the MEMS device based on the geometry of the MEMS sensor.

[0006] Step 2, Modal Analysis and Dominant Mode Selection: Based on the finite element model constructed in Step 1, perform modal analysis, solve the eigenvalue problem to obtain the natural frequencies and corresponding mode shapes of the system, and output the system matrix containing mass and stiffness information and modal contribution data; select the dominant vibration mode based on the modal contribution data; Step 3, Multiphysics Parameter Extraction: For the selected dominant vibration mode, the electrical parameters in the electrical domain, the nonlinear stiffness identification in the mechanical domain, and the damping analysis are performed in parallel. Step 4, Multiphysics Integration and Model Reduction: Integrate the multiphysics parameters extracted in Step 3, and transform the system matrix in Step 1 into a reduced-order model that combines mechanical, electrical and damping effects; Step 5, State Space Description and Encapsulation: Convert the reduced-order model into a state space description suitable for circuit simulators, and encapsulate it using a hardware description language to generate a macro model for system-level co-simulation.

[0007] Furthermore, the equation for solving the eigenvalue problem in step 2 is as follows: in, For the natural frequency, For the quality matrix, Here is the stiffness matrix. This corresponds to the mode shape.

[0008] Furthermore, in step 2, one to three dominant vibration modes are selected based on the modal effective mass ratio, participation factor, orthogonality condition and operational correlation criterion, and the basic transverse mode is selected as the main mode.

[0009] Furthermore, the electrical parameter extraction process in step 3 includes: modeling the capacitance change between electrodes as a function of modal displacement through electrostatic simulation to characterize the charge induction and capacitance sensing mechanisms.

[0010] Furthermore, the nonlinear stiffness identification process in step 3 employs an implicit condensed method to capture geometric nonlinearity; by systematically (or according to a preset step size ratio) changing the amplitude parameters of each dominant mode in finite element static simulation to generate static response data, the stiffness coefficients in the polynomial nonlinear modal force equation are fitted, and the i-th order nonlinear modal force... The approximate expansion is the sum of a quadratic and a cubic polynomial: in, This represents the total number of dominant modes (participating in modal superposition) retained in the reduced-order macromodel. The order index of the dominant mode, with values ​​ranging from 1 to 1. , , , Represents modal coordinates, corresponding to the first modal coordinates, respectively. Rank, number Rank and first The amplitude displacement parameter of the first mode at a specific moment. This represents the second-order (quadratic) polynomial stiffness coefficient, reflecting the... Rank and first Coupling of the first mode to the second The second-order contribution of the first-order modal nonlinear force. : represents the stiffness coefficient of a third-order (cubic) polynomial, reflecting the stiffness of the third-order polynomial. Rank, number Rank and first Coupling of the first mode to the second The third-order contribution of the first-order modal nonlinear force.

[0011] Furthermore, the damping analysis process in step 3 includes: estimating the damping coefficient through computational fluid dynamics (CFD) simulation, and using the quality factor obtained from harmonic analysis. Calculate the damping ratio Based on this, the Rayleigh damping coefficient and quality factor are determined. With damping ratio The relationship is: in, For total energy, This refers to the energy dissipated in each cycle.

[0012] Furthermore, the expression for the order reduction model in step 4 is: in, Representing modal coordinates or modal displacement vectors, it reflects the generalized displacement response state of the structure in the reduced-order modal space. ( The first-order differential, The modal velocity () represents the rate of change of the modal coordinates over time and is directly related to the damping energy dissipation of the system. ( The second derivative, () represents modal acceleration, which is the rate of change of modal velocity over time and is directly related to the inertial force of the system. For the reduced-order quality matrix, The reduced-order damping matrix, For the reduced-order stiffness matrix, the subscripts are... This represents the "reduced order" parameters of the matrix, which are projected onto a lower-dimensional space after dominant mode extraction and multiphysics coupling. This is a nonlinear stiffness force used to capture the geometric nonlinear effects generated in a structure under large deformation conditions; it is a modal displacement. Nonlinear functions. This is electrostatic force, a typical electromechanical coupling term, which depends not only on the applied electrical signal. (Such as AC / DC voltage), and also the transient position of the current moving plate. Closely related. External mechanical force is used to cover other disturbances or inertial excitations imposed by the external environment.

[0013] Furthermore, in step 5, the macro model is encapsulated using the Verilog-A hardware description language and imported into circuit simulation software such as Cadence Virtuoso for system-level co-simulation with the reading and control electronic circuits.

[0014] Furthermore, the MEMS device is a self-compensated resonant electric field microsensor with a driving electrode, a sensing electrode, and a reference electrode; the operating output of this sensor is defined as the ratio of the induced voltage to the reference voltage. in, This represents the effective output voltage, which is the final output result given by the entire sensor system after signal processing and calculation. This value directly reflects the strength of the measured external electrostatic field. This represents the proportional gain constant. This represents the induced voltage, a signal originating from the induction electrode and generated under the principle of charge induction, which includes the external electrostatic field being measured. The information is also affected by the actual vibration amplitude of the structure. This represents the reference voltage, and the signal originates from the reference electrode, which is clamped to a fixed DC voltage. Under these conditions, it only monitors the vibration amplitude and frequency of movable structures, completely unaffected by external electrostatic fields. A system-level co-simulation of frequency tracking and sensitivity temperature drift compensation for the sensor is achieved through seamless integration of a macro-model with a phase-locked loop-based detection circuit.

[0015] The MEMS macro-model construction method based on modal superposition and state-space equations of the present invention has the following advantages: 1. Significantly improved computational efficiency: Compared with full 3D finite element analysis (FEA), the macromodel of this invention achieves a simulation speedup of 423 to 719 times. Memory usage is drastically reduced from approximately 4.2 GB to below 50 MB (a reduction of two orders of magnitude).

[0016] 2. High precision and nonlinear processing: The nonlinear stiffness is handled by combining implicit condensed method, and the error between the frequency response of the macro model and the FEA result is less than 1.6%.

[0017] 3. System-level temperature drift compensation verification: This model can be seamlessly integrated with a phase-locked loop (PLL) detection circuit for transient and sensitivity simulations. Simulations verified the sensor's self-compensation mechanism, effectively suppressing the 22.3% uncompensated drift caused by temperature fluctuations to within ±3%. Attached Figure Description

[0018] Figure 1This is a schematic diagram of the structure of the MEMS resonant electric field microsensor (EFM) in an embodiment of the present invention.

[0019] Figure 2 This is a schematic diagram illustrating the working principle of a MEMS resonant electric field sensor.

[0020] Figure 3 This is a flowchart of the macro model extraction process proposed in this invention.

[0021] Figure 4 This is a block diagram of the EFM macro model and the PLL circuit system-level integration. Detailed Implementation

[0022] To better understand the purpose, structure, and function of this invention, a MEMS macro-model construction method based on modal superposition and state-space equations will be described in further detail below, taking a MEMS resonant electric field sensor as an example and in conjunction with the accompanying drawings.

[0023] A method for constructing MEMS macro-models based on modal superposition and state-space equations includes the following steps: Step 1, Finite Element Modeling: First, a finite element model is constructed based on the geometry of the MEMS resonant electric field sensor. The structure of the MEMS resonant electric field sensor is as follows: Figure 1 As shown, it includes a grounded movable electrode, a driving electrode, a sensing electrode, and a reference electrode. The working principle of a MEMS resonant electric field sensor is as follows: Figure 2 As shown, the grounded movable electrode enters transverse resonance under the electrostatic force of the driving electrode. The driving electrode has a comb-like structure, containing... Each comb tooth has an AC voltage signal applied to it. The driving force is provided to make the movable electrode vibrate, and the sensing electrode measures the external electrostatic field based on the principle of charge induction. The reference electrode potential is clamped to a DC voltage. This forms a comb-like capacitance with the movable structure to monitor vibration amplitude and frequency. The sensor's operating output is defined as the ratio of the induced voltage to the reference voltage: in, This represents the effective output voltage. It is the final output result given by the entire sensor system after signal processing and calculation, and this value directly reflects the strength of the measured external electrostatic field. This represents the proportional gain constant. This represents the induced voltage, a signal originating from the induction electrode and generated under the principle of charge induction, which includes the external electrostatic field being measured. The information is also affected by the actual vibration amplitude of the structure. This indicates the reference voltage. The signal originates from the reference electrode, which is clamped to a fixed DC voltage. It only monitors the vibration amplitude and frequency of movable structures, and is completely unaffected by external electrostatic fields.

[0024] The electrostatic driving force provided by the driving electrodes of the MEMS resonant electric field sensor in, The electrostatic driving force varies with time. The number of comb teeth. The dielectric constant of air is The thickness of the comb tooth structure, The gap between the comb teeth, Here, represents the applied DC bias voltage, and represents the amplitude of the AC drive signal. The angular frequency of the AC drive signal. The time variable is used. The induced current and the reference current are respectively... in, For induced current, This is the conversion coefficient, which is related to the specific electrode structure and sensing area. This is the fundamental resonant angular frequency of the sensor. This represents the modal displacement at the sensing node. These are modal coordinates, reflecting the time-varying behavior of the resonant structure. The transient displacement amplitude at time, The external electrostatic field strength to be measured is denoted as . For reference current, The modal displacement at the reference node. For modal velocity.

[0025] Step 2, Modal Analysis and Dominant Mode Selection: Based on the finite element model constructed in Step 1, solve the eigenvalue problem to obtain the natural frequencies and corresponding mode shapes of the system, and output the system matrix and modal contribution data containing mass and stiffness information; specifically, select 1 to 3 dominant vibration modes according to the modal effective mass ratio, participation factor, orthogonality condition and operational correlation criterion, and select the basic transverse mode as the dominant mode.

[0026] Step 3, Multiphysics Parameter Extraction: For the selected dominant vibration mode, the electrical parameters in the electrical domain, the nonlinear stiffness identification in the mechanical domain, and the damping analysis are performed in parallel. Step 4, Multiphysics Integration and Model Reduction: Integrate the multiphysics parameters extracted in Step 3, and transform the system matrix in Step 1 into a reduced-order model that combines mechanical, electrical and damping effects; Step 5, State Space Description and Encapsulation: Convert the reduced-order model into a state space description suitable for circuit simulators, and encapsulate it using a hardware description language to generate a macro model for system-level co-simulation.

[0027] The equation for solving the eigenvalue problem in step 2 is as follows: in, For the natural frequency, For the quality matrix, Here is the stiffness matrix. This corresponds to the mode shape.

[0028] In step 3, the electrical parameter extraction process includes modeling the capacitance change between electrodes as a function of modal displacement through electrostatic simulation to characterize the charge induction and capacitance sensing mechanisms.

[0029] In step 3, the nonlinear stiffness identification process employs an implicit condensed method to capture geometric nonlinearity. By systematically (or according to a preset step size ratio) changing the amplitude parameters of each dominant mode in finite element static simulation to generate static response data, the stiffness coefficients in the polynomial nonlinear modal force equation are fitted, and the i-th order nonlinear modal force... The approximate expansion is the sum of a quadratic and a cubic polynomial: in, This represents the total number of dominant modes (participating in modal superposition) retained in the reduced-order macromodel. The order index of the dominant mode, with values ​​ranging from 1 to 1. , , , Represents modal coordinates, corresponding to the first modal coordinates, respectively. Rank, number Rank and first The amplitude displacement parameter of the first mode at a specific moment. This represents the second-order (quadratic) polynomial stiffness coefficient, reflecting the... Rank and first Coupling of the first mode to the second The second-order contribution of the first-order modal nonlinear force. : represents the stiffness coefficient of a third-order (cubic) polynomial, reflecting the stiffness of the third-order polynomial. Rank, number Rank and first Coupling of the first mode to the second The third-order contribution of the first-order modal nonlinear force.

[0030] In step 3, the damping analysis process includes estimating the damping coefficient through computational fluid dynamics (CFD) simulation and using the quality factor obtained from harmonic analysis. Calculate the damping ratio Based on this, the Rayleigh damping coefficient and quality factor are determined. With damping ratio The relationship is: in, For total energy, This refers to the energy dissipated in each cycle.

[0031] In step 4, the reduced-order model expression of the system is: in, Representing modal coordinates or modal displacement vectors, it reflects the generalized displacement response state of the structure in the reduced-order modal space. ( The first-order differential, The modal velocity () represents the rate of change of the modal coordinates over time and is directly related to the damping energy dissipation of the system. ( The second derivative, () represents modal acceleration, which is the rate of change of modal velocity over time and is directly related to the inertial force of the system. For the reduced-order quality matrix, The reduced-order damping matrix, For the reduced-order stiffness matrix, the subscripts are... This represents the "reduced order" parameters of the matrix, which are projected onto a lower-dimensional space after dominant mode extraction and multiphysics coupling. This is a nonlinear stiffness force used to capture the geometric nonlinear effects generated in a structure under large deformation conditions; it is a modal displacement. Nonlinear functions. This is electrostatic force, a typical electromechanical coupling term, which depends not only on the applied electrical signal. (Such as AC / DC voltage), and also the transient position of the current moving plate. Closely related. External mechanical force is used to cover other disturbances or inertial excitations imposed by the external environment.

[0032] In step 5, the macro model of the MEMS resonant electric field sensor is encapsulated using the Verilog-A hardware description language and imported into circuit simulation software such as Cadence Virtuoso for comparison. Figure 4The behavioral-level PLL detection circuit shown is connected. Key parameters such as the resonant frequency is set to approximately 2610 Hz and the quality factor to approximately 16. Through co-simulation, the temperature drift of the sensor in the range of -40℃ to 70℃ is evaluated. Simulation results show that the uncompensated output... It will produce the highest The drift. And the compensated output calculated through the macromodel. This verifies that it successfully suppressed drift. Within a certain range, the results are in high agreement with experimental data. Transient response tests show that the phase-locked loop (PLL) performs well within the specified range. Stable locking can be achieved within 2% (error within 2%), further verifying the practicality and extremely high computational efficiency of this macro model in complex circuit interface design.

[0033] This invention has been described through several embodiments. Those skilled in the art will recognize that various changes or equivalent substitutions can be made to these features and embodiments without departing from the spirit and scope of the invention. Furthermore, under the teachings of this invention, modifications can be made to these features and embodiments to adapt to specific situations and materials without departing from the spirit and scope of the invention. Therefore, this invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are protected by this invention.

Claims

1. A method for constructing MEMS macro-models based on modal superposition and state-space equations, characterized in that, Includes the following steps: Step 1, Finite Element Modeling: Construct a finite element model of the MEMS device based on the geometry of the MEMS sensor; Step 2, Modal Analysis and Dominant Mode Selection: Based on the finite element model constructed in Step 1, perform modal analysis, solve the eigenvalue problem to obtain the natural frequencies and corresponding mode shapes of the system, and output the system matrix containing mass and stiffness information and modal contribution data; select the dominant vibration mode based on the modal contribution data; Step 3, Multiphysics Parameter Extraction: For the selected dominant vibration mode, the electrical parameters in the electrical domain, the nonlinear stiffness identification in the mechanical domain, and the damping analysis are performed in parallel. Step 4, Multiphysics Integration and Model Reduction: Integrate the multiphysics parameters extracted in Step 3, and transform the system matrix in Step 1 into a reduced-order model that combines mechanical, electrical and damping effects; Step 5, State Space Description and Encapsulation: The reduced-order model is converted into a state space description suitable for circuit simulators, and encapsulated using a hardware description language to generate a macro model for system-level co-simulation.

2. The MEMS macro-model construction method based on modal superposition and state-space equations according to claim 1, characterized in that, The equation for solving the eigenvalue problem in step 2 is as follows: ； in, For the natural frequency, For the quality matrix, Here is the stiffness matrix. This corresponds to the mode shape.

3. The MEMS macro-model construction method based on modal superposition and state-space equations according to claim 1, characterized in that, In step 2, one to three dominant vibration modes are selected based on the modal effective mass ratio, participation factor, orthogonality condition and operational correlation criterion, and the basic transverse mode is selected as the dominant mode.

4. The MEMS macro-model construction method based on modal superposition and state-space equations according to claim 1, characterized in that, The electrical parameter extraction process in step 3 includes: modeling the capacitance change between electrodes as a function of modal displacement through electrostatic simulation to characterize the charge induction and capacitance sensing mechanisms.

5. The MEMS macro-model construction method based on modal superposition and state-space equations according to claim 1, characterized in that, The nonlinear stiffness identification process in step 3 employs an implicit condensed method to capture geometric nonlinearity. By systematically changing the amplitude parameters of each dominant mode in finite element static simulation to generate static response data, the stiffness coefficients in the polynomial nonlinear modal force equation are fitted, and the i-th order nonlinear modal force... The approximate expansion is the sum of a quadratic and a cubic polynomial: ； in, This represents the total number of dominant modes retained in the reduced-order macromodel. The order index of the dominant mode, with values ​​ranging from 1 to 1. , , , Represents modal coordinates, corresponding to the first modal coordinates, respectively. Rank, number Rank and first The amplitude displacement parameter of the first mode at a specific moment; This represents the stiffness coefficient of a quadratic polynomial, reflecting the... Rank and first Coupling of the first mode to the second The second-order contribution of the first-order modal nonlinear force; This represents the stiffness coefficient of a cubic polynomial, reflecting the... Rank, number Rank and first Coupling of the first mode to the second The third-order contribution of the first-order modal nonlinear force.

6. The MEMS macro-model construction method based on modal superposition and state-space equations according to claim 1, characterized in that, The damping analysis process in step 3 includes: estimating the damping coefficient through computational fluid dynamics simulation, and obtaining the quality factor using harmonic analysis. Calculate the damping ratio And based on this, the Rayleigh damping coefficient is determined; the quality factor With damping ratio The relationship is: ； in, For total energy, This refers to the energy dissipated in each cycle.

7. The MEMS macro-model construction method based on modal superposition and state-space equations according to claim 1, characterized in that, The expression for the order reduction model in step 4 is: ； in, Represents modal coordinates or modal displacement vectors; Indicates modal velocity; Indicates modal acceleration; For the reduced-order quality matrix, The reduced-order damping matrix, For the reduced-order stiffness matrix, the subscripts are... This means that the matrix is ​​a reduced-order parameter projected onto a low-dimensional space after dominant mode extraction and multi-physics coupling; It is a nonlinear stiffness force; It is electrostatic force; External mechanical force.

8. The MEMS macro-model construction method based on modal superposition and state-space equations according to claim 1, characterized in that, In step 5, the macro model is encapsulated using Verilog-A hardware description language and imported into Cadence Virtuoso circuit simulation software for system-level co-simulation with the reading and control electronic circuits.

9. The MEMS macro-model construction method based on modal superposition and state-space equations according to claim 1, characterized in that, The MEMS device is a self-compensated resonant electric field microsensor with a driving electrode, a sensing electrode, and a reference electrode; the operating output of the sensor is defined as the ratio of the induced voltage to the reference voltage. ； in, Indicates the effective output voltage; Represents the proportional gain constant; Indicates induced voltage; The reference voltage is represented by the macro model. The macro model is seamlessly integrated with the phase-locked loop-based detection circuit to perform system-level co-simulation of frequency tracking and sensitivity temperature drift compensation for the sensor.

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