A nonlinear fitting method and system of impedance equivalent model

Abstract

The application discloses a nonlinear fitting method and system of an impedance equivalent model, voltage and current data are collected at configured frequency points through a frequency sweeping hardware system, corresponding impedance values are obtained through orthogonal demodulation for the data at each frequency, a matched impedance model is selected from a plurality of preset impedance equivalent circuits, a plurality of initial model parameters are predicted through a multi-physical field coupling initial value prediction algorithm, the initial model parameters are combined with a parameter physical constraint optimization strategy, and an optimal solution is obtained through nonlinear least square fitting, so that the algorithm has the advantages of accelerating the convergence speed and reducing the requirement for computing power, model parameters are automatically adjusted in the algorithm operation process, errors between model prediction values and actual measurement values are minimized, and the model parameters are continuously adjusted, so that accurate fitting of the equivalent model is realized.

Description

Technical Field

[0001] This invention belongs to the field of electronic measurement and electrochemical analysis technology. Specifically, it relates to a nonlinear fitting method and system for impedance equivalent models of circuit components. Background Technology

[0002] Impedance equivalent model analysis technology is widely used in electronic circuit design and electrochemical system monitoring. Its core objective is to characterize the impedance characteristics of complex systems through mathematical models or equivalent circuits.

[0003] Establishing a complete and accurate impedance equivalent model is of great significance for circuit signal integrity analysis, thermoelectric coupling of power devices, parasitic parameter extraction, and battery performance analysis. Currently, common methods for impedance equivalent model analysis include time-domain analysis and frequency response analysis. Time-domain analysis extracts impedance information by analyzing the system's response in the time domain; frequency response analysis extracts impedance information by measuring the system's response at different frequencies.

[0004] Current impedance equivalent model analysis techniques have the following problems:

[0005] (1) Reliance on human experience and subjective judgment: Traditional methods such as manually fitting Nyquist plots and Bode plots require inferring the equivalent circuit structure through the shape of the data curve (such as a semicircle or a straight line segment). The process relies on the engineer's experience, is inefficient, and is easily affected by subjective factors.

[0006] (2) The physical meaning of model parameters is ambiguous: Although some mathematical fitting methods (such as polynomial fitting) can match data curves, the parameters lack actual physical meaning and cannot correspond to the characteristics of real components such as resistors, inductors, and capacitors. It is difficult to combine domain knowledge to explain the system mechanism.

[0007] (3) Insufficient fitting ability for complex systems: For complex systems containing multi-order RC networks, distributed parameters or nonlinear elements (such as constant phase element CPE), traditional methods are difficult to optimize coupling parameters simultaneously.

[0008] (4) Limited engineering application value: Existing methods mostly remain at the level of data visualization, requiring data to be exported for processing. The fitting results cannot be directly transferred to engineering scenarios such as algorithm design, making it difficult to meet the actual system modeling and optimization needs. Summary of the Invention

[0009] To address the aforementioned problems, this invention proposes a nonlinear fitting method and system for impedance equivalent models. It combines frequency scanning technology with a pre-defined physical equivalent model circuit, utilizing nonlinear least squares algorithms, adaptive model selection algorithms, and initial value prediction algorithms to improve the automation level of impedance model fitting. This solves the problems of existing methods, such as heavy reliance on manual intervention, poor physical interpretability of the model, insufficient fitting accuracy for complex systems, and high computational requirements. Through deep collaborative optimization of hardware, algorithms, and display, it achieves millisecond-level modeling of complex systems, providing an efficient, accurate, and engineering-practical technical solution for impedance characteristic analysis.

[0010] The present invention is implemented using the following technical solutions:

[0011] A nonlinear fitting method for the impedance equivalent model is proposed, including:

[0012] S1, acquire measurement signals at each frequency point based on the configured frequency sweep parameters;

[0013] S2, calculate the impedance value at each frequency and select the appropriate impedance model;

[0014] S3, according to the selected impedance model, invoke the multiphysics coupling prediction algorithm to predict the initial model parameters of the impedance model; the multiphysics coupling prediction algorithm includes:

[0015] For circuit component models, the initial model parameters are solved by a step-by-step derivation method based on the characteristics of the resonant point, the characteristics of the minimum frequency point, and / or the reconstructed impedance equation.

[0016] For the electrochemical model, based on the impedance spectrum of the model, the center and radius of the circle or arc in the Nyquist plot are calculated, and the initial model parameters are solved based on the center and radius data.

[0017] S4. Based on the physical constraint optimization strategy for model construction parameters, the initial model parameters are used as the initial solution. The nonlinear least squares method is called to iteratively adjust the model parameters to obtain the optimal solution.

[0018] In some embodiments of the present invention, in step S2, an adaptive model selection algorithm is called to match the impedance model for the circuit component model;

[0019] The adaptive model selection algorithm includes:

[0020] Calculate the initial phase angle ;

[0021] The maximum and minimum points of the amplitude are detected, and the model is matched using the total number of extreme points N and the initial phase angle.

[0022] ;

[0023] As described above, impedance model A is the equivalent circuit model of a resistor, capacitor, and inductor connected in parallel; impedance model B is the equivalent circuit of an inductor and resistor connected in series and then connected in parallel with a capacitor; impedance model C is the equivalent circuit of a resistor, capacitor, and inductor connected in series; impedance model D is the equivalent circuit of an inductor connected in series with a parallel capacitor and resistor; and impedance model E is the equivalent circuit of capacitor one connected in parallel with an inductor, capacitor two, and resistor connected in series.

[0024] In some embodiments of the present invention, in S3, for the circuit component model,

[0025] (1) Solve for the initial model parameters based on the characteristics of the resonance point; including:

[0026] Based on the maximum amplitude corresponding to the series resonant point and the minimum amplitude corresponding to the parallel resonant point, the resonant point is obtained by extreme value analysis of the sweep frequency data amplitude. The resonant point frequency ω_res is substituted into the imaginary part of the impedance function to establish the equation Im(Z)(ω_res)=0, and the initial model parameters are obtained by solving it.

[0027] (2) Solve for the initial model parameters based on the characteristics of the minimum frequency point; including:

[0028] For cases where a capacitor is connected in parallel or an inductor is connected in series in a composite circuit, a simplified model is obtained by treating the parallel capacitor as an open circuit and the series inductor as a short circuit under low-frequency conditions. The minimum frequency ω_min is substituted into the real and imaginary parts of the simplified model, and the real and imaginary part equations are established with the measured data respectively. The initial model parameters are obtained by solving the equations.

[0029] (3) For models where the inductance cannot be directly determined from the characteristics of the resonant point and the minimum frequency point, the initial model parameters are solved by step-by-step calculation; including:

[0030] Based on the determined initial model parameters, the impedance equation is reconstructed, and the explicit expression for the inductance parameter L is derived; the inductance estimate at each frequency point is solved by traversing the swept frequency data samples. ;

[0031] The final initial value L_init = mean is obtained by using the arithmetic mean method. ).

[0032] In some embodiments of the present invention, in S3, for the electrochemical model, a Nyquist plot of the model impedance spectrum is established, and the center and radius of the circles or arcs in the plot are calculated, including:

[0033] For composite components connected in parallel with RC circuits:

[0034] ;

[0035] , ;

[0036] The coordinates of the center of the circle are (R / 2, 0), and the radius is R / 2.

[0037] For composite components with R and CPE connected in parallel:

[0038] ,

[0039] ,

[0040] ,

[0041] The coordinates of the center and the radius are obtained as follows:

[0042] , ;in, The equivalent capacitance of the load-transfer structure. These are the load-transfer structural characteristic values; R is the resistance value, and C is the capacitance value. For current / voltage frequency, The real part of the impedance. This represents the imaginary part of the impedance.

[0043] The initial model parameters are solved based on the center and radius data, including:

[0044] Initial resistance value: On the Nyquist plot, an interpolation algorithm is used to determine a value that intersects the horizontal axis of the real part as the initial value of RS. ;

[0045] Initial inductance value: Solved using the imaginary part of the highest frequency point, it is equal to the imaginary part of the impedance at the highest frequency point divided by the corresponding angular frequency;

[0046] Initial values ​​of parallel composite elements: Calculate the initial resistance value. The real part of the impedance at the maximum point peak_index;

[0047] by Calculate the initial value of the capacitor. The frequency of the maximum point peak_index;

[0048] For the R-CPE parallel circuit, based on the imaginary part and frequency of the maximum point of the irregular circular arc segment, as well as the center and radius of the circle, the initial values ​​of α and Q are further obtained. , Where peak_index is the maximum point of the negative imaginary part. The initial value of the resistance. ; This is the imaginary part of the impedance at the point of maximum value.

[0049] A nonlinear fitting system based on an impedance equivalent model is proposed, comprising:

[0050] The frequency sweeping hardware module is used to acquire measurement signals at each frequency point based on the configured frequency sweeping parameters;

[0051] The data processing module is used to calculate the impedance values ​​at various frequencies and select an appropriate impedance model based on the device under test. According to the selected impedance model, it calls a multiphysics coupling prediction algorithm to predict the initial model parameters of the impedance model. Based on the physical constraint optimization strategy for model construction parameters, using the initial model parameters as the initial solution, it calls a nonlinear least squares method to iteratively adjust the model parameters to obtain the optimal solution. The multiphysics coupling prediction algorithm includes: for circuit component models, solving for the initial model parameters using a step-by-step derivation method based on resonant point characteristics, minimum frequency point characteristics, and / or reconstructed impedance equations; for electrochemical models, calculating the center and radius of the circles or arcs in the Nyquist plot based on the model's impedance spectrum, and solving for the initial model parameters based on the center and radius data.

[0052] The display module allows for frequency sweep configuration via a UI interface, and displays the fitting results and responses in the form of Nyquist plots or Bode plots.

[0053] In some embodiments of the present invention, the data processing module calls an adaptive model selection algorithm to match the impedance model for the circuit component model;

[0054] The adaptive model selection algorithm includes:

[0055] Calculate the initial phase angle ;

[0056] The maximum and minimum points of the amplitude are detected, and the model is matched using the total number of extreme points N and the initial phase angle.

[0057] ;

[0058] As described above, impedance model A is the equivalent circuit model of a resistor, capacitor, and inductor connected in parallel; impedance model B is the equivalent circuit of an inductor and resistor connected in series and then connected in parallel with a capacitor; impedance model C is the equivalent circuit of a resistor, capacitor, and inductor connected in series; impedance model D is the equivalent circuit of an inductor connected in series with a parallel capacitor and resistor; and impedance model E is the equivalent circuit of capacitor one connected in parallel with an inductor, capacitor two, and resistor connected in series.

[0059] In some embodiments of the present invention, the data processing module is designed for circuit component models.

[0060] (1) Solve for the initial model parameters based on the characteristics of the resonance point; including:

[0061] Based on the maximum amplitude corresponding to the series resonant point and the minimum amplitude corresponding to the parallel resonant point, the resonant point is obtained by extreme value analysis of the sweep frequency data amplitude. The resonant point frequency ω_res is substituted into the imaginary part of the impedance function to establish the equation Im(Z)(ω_res)=0, and the initial model parameters are obtained by solving it.

[0062] (2) Solve for the initial model parameters based on the characteristics of the minimum frequency point; including:

[0063] For cases where a capacitor is connected in parallel or an inductor is connected in series in a composite circuit, a simplified model is obtained by treating the parallel capacitor as an open circuit and the series inductor as a short circuit under low-frequency conditions. The minimum frequency ω_min is substituted into the real and imaginary parts of the simplified model, and the real and imaginary part equations are established with the measured data respectively. The initial model parameters are obtained by solving the equations.

[0064] (3) For models where the inductance cannot be directly determined from the characteristics of the resonant point and the minimum frequency point, the initial model parameters are solved by step-by-step calculation; including:

[0065] Based on the determined initial model parameters, the impedance equation is reconstructed, and the explicit expression for the inductance parameter L is derived; the inductance estimate at each frequency point is solved by traversing the swept frequency data samples. ;

[0066] The final initial value L_init = mean is obtained by using the arithmetic mean method. ).

[0067] In some embodiments of the present invention, the data processing module, for an electrochemical model, establishes a Nyquist plot of the model impedance spectrum and calculates the center and radius of the circles or arcs in the plot, including:

[0068] For composite components connected in parallel with RC circuits:

[0069] ;

[0070] , ;

[0071] The coordinates of the center of the circle are (R / 2, 0), and the radius is R / 2.

[0072] For composite components with R and CPE connected in parallel:

[0073] ,

[0074] ,

[0075] ,

[0076] The coordinates of the center and the radius are obtained as follows:

[0077] , ;in, The equivalent capacitance of the load-transfer structure. These are the load-transfer structural characteristic values; R is the resistance value, and C is the capacitance value. For current / voltage frequency, The real part of the impedance. This represents the imaginary part of the impedance.

[0078] The initial model parameters are then solved based on the center and radius data, including:

[0079] Initial resistance value: On the Nyquist plot, an interpolation algorithm is used to determine a value that intersects the horizontal axis of the real part as the initial value of RS. ;

[0080] Initial inductance value: Solved using the imaginary part of the highest frequency point, it is equal to the imaginary part of the impedance at the highest frequency point divided by the corresponding angular frequency;

[0081] Initial values ​​of parallel composite elements: Calculate the initial resistance value. The real part of the impedance at the maximum point peak_index;

[0082] by Calculate the initial value of the capacitor. The frequency of the maximum point peak_index;

[0083] For the R-CPE parallel circuit, based on the imaginary part and frequency of the maximum point of the irregular circular arc segment, as well as the center and radius of the circle, the values ​​of α and Q are further obtained. , Where peak_index is the maximum point of the negative imaginary part. The initial value of the resistance. ; This is the imaginary part of the impedance at the point of maximum value.

[0084] Compared with the prior art, the advantages and positive effects of the present invention are as follows: In the nonlinear fitting method and system for the impedance equivalent model proposed in this invention, voltage and current data are collected at configured frequency points through a frequency sweeping hardware system. For the data at each frequency, the corresponding impedance values ​​are obtained through orthogonal demodulation. A matching impedance model is selected from a variety of preset impedance equivalent circuits. A set of initial model parameters is predicted using a multi-physics coupling initial value prediction algorithm. The initial model parameters are combined with a parameter physical constraint optimization strategy, and the optimal solution is obtained by fitting using a nonlinear least squares method. This method has the advantages of accelerating the convergence speed of the algorithm and reducing the requirements for computing power. The model parameters are automatically adjusted during the algorithm operation to minimize the error between the model prediction value and the actual measurement value. By continuously adjusting the model parameters, accurate fitting of the equivalent model is achieved.

[0085] This invention constructs a system that combines a physical model with nonlinear least squares fitting. By deeply coupling the adaptive model selection algorithm with the nonlinear least squares optimization framework, it greatly improves computational efficiency, reduces manual intervention, and ensures that the parameters of each component have clear physical meaning. Ultimately, it provides an impedance analysis method that combines high automation and strong interpretability.

[0086] This invention proposes an initial value prediction algorithm for model parameters under multi-physics coupling, constructing an optimization strategy under parameter physical constraint space. By associating the collected frequency domain voltage and current data with the equivalent circuit topology, and by analyzing resonance peaks, phase inflection points, etc., the initial model parameters are predicted and constrained. This significantly reduces the number of algorithm iterations, avoids local optima, improves the convergence success rate, reduces the dependence on computing power, and greatly improves the fitting speed and accuracy for complex systems. It can accurately fit complex systems including multi-order RC networks and constant-phase elements. At the same time, it uses an ARM embedded chip to complete the entire process from data acquisition, orthogonal demodulation to parameter optimization locally, meeting engineering requirements and facilitating engineering applications.

[0087] Other features and advantages of the present invention will become clearer after reading the detailed description of the embodiments of the present invention in conjunction with the accompanying drawings. Attached Figure Description

[0088] The accompanying drawings, as part of this invention, are provided to further illustrate the invention. The illustrative embodiments and descriptions are used to explain the invention but do not constitute an undue limitation thereof. Clearly, the drawings described below are merely some embodiments; those skilled in the art can obtain other drawings based on these drawings without creative effort.

[0089] Figure 1 This is the overall framework of the nonlinear fitting system based on the impedance equivalent model proposed in this invention;

[0090] Figure 2 This diagram illustrates the execution steps of the nonlinear fitting method for the impedance equivalent model proposed in this invention.

[0091] Figure 3 This is a schematic diagram of the impedance model type of the circuit components given in the embodiments of the present invention;

[0092] Figure 4 For the purposes of this embodiment of the invention Figure 3 The fitting effect of model B shown;

[0093] Figure 5 This is a schematic diagram of the type of electrochemical model given in the embodiments of the present invention;

[0094] Figure 6 For the purposes of this embodiment of the invention Figure 5 The impedance spectrum of model C is shown.

[0095] It should be noted that these accompanying drawings and textual descriptions are not intended to limit the scope of the invention in any way, but rather to illustrate the concept of the invention to those skilled in the art by referring to specific embodiments. Detailed Implementation

[0096] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments will be clearly and completely described below with reference to the accompanying drawings. The following embodiments are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0097] This invention addresses the key technical bottlenecks in traditional impedance equivalent model analysis, such as low operating efficiency, insufficient fitting ability for complex systems, and difficulties in engineering implementation. It proposes a modeling solution based on the optimized nonlinear least-squares method, which mainly includes: acquiring a frequency sweep signal through a frequency sweep hardware module, combining it with a preset physical equivalent circuit, and analyzing the equivalent model through a nonlinear least-squares fitting algorithm. The frequency sweeping hardware module is mainly controlled by an FPGA and interacts with an ARM embedded chip (data processing module). It receives configuration information sent by the ARM and sends the collected voltage and current data to the ARM for data processing. In the ARM, the corresponding impedance values ​​are obtained through digital quadrature demodulation for the frequency response data at each frequency. The ARM has preset multiple impedance equivalent circuits, including circuit component models and electrochemical models. The optimal model is automatically identified and matched based on user selection or through an adaptive model selection algorithm. The Levenberg-Marquardt algorithm is used to achieve nonlinear least squares fitting. Combined with multi-physics coupling treatment prediction algorithm and parameter physical constraint optimization strategy, the convergence speed of the algorithm is accelerated and the computing power requirement is reduced. The model parameters are automatically adjusted during the algorithm operation to minimize the error between the model prediction value and the actual measurement value. By continuously adjusting the model parameters, the equivalent model is accurately fitted. After the fitting is completed, it is displayed in the display module.

[0098] like Figure 1 The diagram illustrates the interactions between the various parts of the system. The entire nonlinear fitting system for the impedance equivalent model is divided into three parts: a frequency sweeping hardware module controlled by an FPGA, a data processing module controlled by an ARM, and a display module for human-computer interaction. Users can configure the frequency sweeping through the UI interface of the display unit. The ARM processes the data and sends it to the FPGA. The FPGA collects voltage and current information at each frequency point according to the corresponding settings and sends it to the ARM. The nonlinear fitting method of this invention processes the data, and the fitted results, along with the corresponding Nyquist or Bode plots, are displayed on the display module.

[0099] The implementation process of nonlinear fitting of the impedance equivalent model is as follows: Figure 2 As shown, it includes:

[0100] S1: Acquire measurement signals at each frequency point based on the configured sweep frequency parameters.

[0101] 1. Configure the frequency sweep parameters in the UI interface of the display module.

[0102] The frequency sweep parameters include the number of sweep points and the measurement signal. The set parameters are preprocessed by the ARM and then sent to the FPGA.

[0103] 2. The FPGA acquires the measurement signals at each frequency point.

[0104] The frequency sweep circuit is configured by FPGA to collect voltage and current at each frequency point and transmit the collected data back to ARM in real time.

[0105] S2: Calculate the impedance value at each frequency and select the appropriate impedance model.

[0106] 1. The ARM receives voltage and current data at various frequency points, performs quadrature demodulation, removes harmonics and noise, and calculates the impedance value at the corresponding frequency.

[0107] Specifically, orthogonal reference signals are first generated using a lookup table method or real-time calculation:

[0108] ,

[0109] ;

[0110] Then, the acquired signal is multiplied by the orthogonal signal and integrated to obtain the orthogonal components:

[0111] , ;

[0112] By performing the above processing on the voltage and current respectively, the corresponding amplitude and phase are calculated, and finally the impedance value is obtained:

[0113] , ;

[0114] , ;

[0115] , .

[0116] 2. Based on the frequency and impedance value, call the adaptive model selection algorithm to determine the appropriate impedance model.

[0117] Impedance models include electrochemical models and circuit component models.

[0118] The appropriate impedance model can be manually selected in the UI interface. The circuit component model can automatically adapt to the impedance model by calling the adaptive model selection algorithm given in this invention.

[0119] Specifically, first calculate the initial phase angle: ;

[0120] Next, the maximum (peak) and minimum (trough) points of the amplitude are detected. The total number of extreme points N = the number of peaks + the number of troughs. Finally, a suitable model is automatically matched based on the number of extreme points and the initial phase angle.

[0121] .

[0122] The above, combined with Figure 3 As shown, impedance model A is the equivalent circuit model of a resistor, capacitor, and inductor connected in parallel; impedance model B is the equivalent circuit of an inductor and resistor connected in series and then connected in parallel with a capacitor; impedance model C is the equivalent circuit of a resistor, capacitor, and inductor connected in series; impedance model D is the equivalent circuit of an inductor connected in series with a parallel capacitor and resistor; and impedance model E is the equivalent circuit of capacitor one connected in parallel with an inductor, capacitor two, and resistor connected in series.

[0123] Then, the interface settings and raw calculation data are encapsulated into JSON data blocks and processed by a nonlinear least squares algorithm.

[0124] S3: Based on the selected impedance model, call the multiphysics coupling initial value prediction algorithm to predict the initial model parameters of the impedance model.

[0125] The JSON data block is parsed, and according to the selected impedance model, the multiphysics coupling initial value prediction algorithm is called to predict a set of initial values ​​of the initial model parameters as the initial solution of the nonlinear least squares algorithm.

[0126] In this invention, the multiphysics coupling initial value prediction algorithm employs different calculation strategies for circuit component models and electrochemical models.

[0127] For example Figure 3 The circuit component model shown is used to solve for the initial model parameters from the following three dimensions through frequency sweep response characteristic analysis:

[0128] (1) Solve for the initial value based on the characteristics of the resonance point.

[0129] When the system is in resonance, the circuit exhibits purely resistive characteristics (Im(Z)=0). The series resonance point corresponds to the maximum amplitude, and the parallel resonance point corresponds to the minimum amplitude. Therefore, the resonance point can be obtained by performing extreme value analysis on the amplitude of the swept frequency data. By expanding the impedance function into an explicit expression of the real part Re(Z) and the imaginary part Im(Z), and substituting the ω_res corresponding to the resonance frequency point into the imaginary part equation Im(Z)(ω_res)=0, the corresponding constraint equation can be established.

[0130] (2) Solve for the initial value based on the characteristics of the minimum frequency point.

[0131] For cases where a capacitor is connected in parallel to a composite circuit or an inductor is connected in series to a composite circuit, such as... Figure 3The typical topologies shown in Model B (RL series) and Model D (RC parallel) can be simplified under low-frequency conditions by treating the parallel capacitor as an open circuit and the series inductor as a short circuit. The minimum frequency corresponding to ω_min is substituted into the real part Re(Z)(ω_min) and imaginary part Im(Z)(ω_min) expressions of the simplified model. The real part equation and imaginary part equation are established with the measured data respectively, forming a double constraint.

[0132] (3) Solve for the initial values ​​step by step.

[0133] For models where the inductance cannot be directly solved using conventional feature points, a step-by-step derivation strategy for the reconstructed impedance equation is adopted:

[0134] a. Based on the determined initial model parameters, reconstruct the impedance equation and derive the explicit expression for the inductance parameter L;

[0135] b. Traverse the swept frequency data samples to solve for the inductance estimate at each frequency point. ;

[0136] c. Obtain the final initial inductance value L_init = mean( using the arithmetic mean method). ).

[0137] Taking model B as an example, the fitting effect is as follows: Figure 4 As shown, at low frequencies, C is equivalent to an open circuit, at the minimum frequency point. Corresponding impedance value Approximate to At this time, the initial resistance value Initial inductance value ;

[0138] The model is then converted to the following form: ;

[0139] Resonance point frequency Since the imaginary part is 0 at this point, the initial value of the capacitor can be obtained:

[0140] .

[0141] In this invention, a dedicated prediction module is constructed for the electrochemical model based on its unique physical field coupling mechanism. A typical impedance circuit is shown below. Figure 5 As shown; electrochemical impedance spectroscopy is often represented by Nyquist plots, such as... Figure 6The impedance spectrum of model C is shown. In the Nyquist plot, the larger the radius of the circle or arc, the greater the impedance. The presence of multiple arcs indicates the existence of multiple time constants in the electrochemical system. One time constant represents one state variable, usually a segment of capacitive arc characteristics, indicating that there is only one composite element in the equivalent circuit, as shown in models A and B. Two time constants represent two state variables, usually two segments of capacitive arc characteristics at different frequencies, and the equivalent circuit is a series or parallel connection of two composite elements, as shown in models C and D. Initial values ​​can be calculated based on these characteristics.

[0142] The center and radius of the arc are important bases for calculating initial values. For RC parallel composite components, the process of determining the center and radius is as follows:

[0143] , , ;

[0144] The equation of the circle's trajectory is satisfied, so the center coordinates are (R / 2, 0) and the radius is R / 2.

[0145] Another type of composite element is the parallel connection of R and CPE (constant-phase element). The constant-phase element is a non-intuitive equivalent circuit element used to describe the "diffuse effect," meaning that due to the inhomogeneity of the electrode or system's internal characteristics, the response characteristics may shift to varying degrees. CPE is often represented by the equivalent capacitance Q of the charge transfer structure and the characteristic value α of the charge transfer structure. The process for determining the center and radius of the parallel R and CPE circuit is as follows:

[0146] ;

[0147] , ;

[0148] Then the coordinates of the center and the radius are:

[0149] , .

[0150] With the center and radius determined, the initial model parameter values ​​can be calculated. The specific process is as follows:

[0151] (1) Determine the initial value of the resistance.

[0152] When the frequency reaches a certain value, the capacitance decreases, the imaginary impedance has a lower impact, and the real impedance becomes the dominant impedance. At this point, the impedance can be equivalent to a resistor. On the Nyquist plot, this is represented by the intersection point with the horizontal axis of the real part. Since the frequency sweep points are discrete, an interpolation algorithm can be used to approximately determine a value at the intersection point with the horizontal axis of the real part as the initial value of RS. .

[0153] (2) Determine the initial value of the inductance.

[0154] The inductance L1 is solved using the imaginary part of the impedance at the highest frequency. In the high-frequency region, the inductive reactance of the inductor dominates the imaginary part of the impedance, and the effects of the series resistance and parallel capacitance can be ignored. Therefore, the initial value of the inductance L1 is equal to the imaginary part of the impedance at the highest frequency divided by the corresponding angular frequency.

[0155] (3) Determine the initial values ​​of the parallel composite elements.

[0156] In both RC parallel and R-CPE parallel circuits, the real part coordinate of the center is R / 2. However, due to the presence of RS, in practical applications, the real part coordinate should be RS+R / 2. Finding the peak_index, the maximum point of the negative imaginary part, gives the corresponding real part impedance as the real part coordinate. The initial resistance value in the parallel composite element is then calculated as follows:

[0157] ; This is the real part of the impedance at the maximum point peak_index.

[0158] For RC parallel circuits, some quantities actually exhibit the following relationship at their maximum points: The initial value of the capacitor can be obtained: , The frequency of the maximum point peak_index.

[0159] For the R-CPE parallel circuit, based on the imaginary part and frequency of the maximum point of the irregular circular arc segment, as well as the center and radius of the circle, the initial values ​​of α and Q are further obtained:

[0160] , ;

[0161] .

[0162] by Figure 5 Taking model C as an example, the fitting results are as follows: Figure 6 As shown, first, the initial value of RS is calculated. By detecting the zero-crossing point of the imaginary impedance (when the imaginary part changes from negative to positive), the value of the real part at the point where the imaginary part equals 0 is estimated using linear interpolation. If there is no zero-crossing point, the minimum value of the real part is taken. The specific formula is as follows:

[0163] ;in, This is an estimated impedance value. This is the actual impedance value.

[0164] L1 initial value uses the highest frequency point The calculation is based on the principle dominated by inductive reactance ωL, and the formula is as follows:

[0165] ,in, This is the maximum estimated impedance value.

[0166] The initial value calculation of R1 first finds the trough of the imaginary impedance, and sets it as follows: The specific formula is as follows:

[0167] ;

[0168] Then alpha1_init (CPE1 parameter) can be calculated using the following formula:

[0169] ;

[0170] Q1_init (CPE1 coefficient) is calculated as follows:

[0171] ;

[0172] The initial value of R² can be determined by multiplying the difference between the x-coordinates of the second maximum and minimum points by 2, based on the center of the circle.

[0173] ;

[0174] The calculation principle of alpha2_init (CPE2 exponent) is similar to that of alpha1_init, except that the extreme point is chosen differently:

[0175] ;

[0176] The formula for calculating Q2_init (CPE2 coefficient) is as follows:

[0177] .

[0178] S4: Based on the physical constraint optimization strategy for model construction parameters, using the initial model parameters as the initial solution, call the nonlinear least squares algorithm to iteratively adjust the model parameters and obtain the optimal solution.

[0179] While solving for the initial solution of the algorithm, it is also necessary to construct a parameter physical constraint optimization strategy based on the model. Since there are many models, in order to ensure the accuracy of the fitting, it is generally chosen to set the parameter boundaries of all resistors in the model to be greater than 0. This set of constraints and initial solutions can reduce the number of algorithm iterations, speed up the convergence speed of the algorithm, and improve the fitting accuracy of the algorithm.

[0180] The initial solution is passed to Levenberg-Marquardt for iteration, which continuously reduces the error between the observed and actual values ​​until the optimal solution is obtained, i.e., the parameters of each impedance model. The impedance values ​​of each fitted point are calculated based on the set frequency points and the fitted parameters. The chi-square test is performed to evaluate the fitting effect. The chi-square value must be less than 0.1%. After the test is passed, the fitting results are packaged into JSON data blocks.

[0181] S5: Display the Nyquist plot or Bode plot of the fitting results and response on the display module.

[0182] The JSON data block is parsed using the Qt framework. The fitted impedance point data is converted into QVariantMap type data, and data cleaning is performed to eliminate outliers. The amplitude and phase are calculated, and the electrochemical model is displayed in the form of a Nyquist plot to show the charge transfer process and diffusion control characteristics at high and low frequencies. The circuit component model is displayed in the form of a Bode plot to show the frequency response and capacitive inductance characteristics. Figure 6 The comparison of the fitting performance of the models shown is presented in the form of Nyquist plots, demonstrating that the algorithm still has a good fitting effect on complex models.

[0183] It should be noted that, in the specific implementation process, the control part mentioned above is implemented by the FPGA executing computer-executed instructions in software form stored in the memory. This will not be elaborated here. The programs corresponding to the actions performed by the control circuit can all be stored in software form in the FPGA-readable storage medium of the system, so that the FPGA can call and execute the corresponding operations of the above modules.

[0184] The computer-readable storage media mentioned above may include volatile memory, such as random access memory; may also include non-volatile memory, such as read-only memory, flash memory, hard disk or solid-state drive; and may also include combinations of the above types of memory.

[0185] The term "processor" as mentioned above can also refer to a collective of multiple processing elements. For example, a processor can be a central processing unit (CPU), or other general-purpose processors, digital signal processors, application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs) or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. A general-purpose processor can be a microprocessor or any conventional processor, and it can also be a special-purpose processor.

[0186] It should be noted that the above description is not intended to limit the present invention, and the present invention is not limited to the examples given above. Any changes, modifications, additions or substitutions made by those skilled in the art within the scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A nonlinear fitting method for an impedance equivalent model, characterized in that, include: S1, acquire measurement signals at each frequency point based on the configured frequency sweep parameters; S2, calculate the impedance value at each frequency and select the appropriate impedance model; S3, according to the selected impedance model, call the multiphysics coupling prediction algorithm to predict the initial model parameters of the impedance model; The multiphysics coupling prediction algorithm includes: For circuit component models, the initial model parameters are solved by a step-by-step derivation method based on the characteristics of the resonant point, the characteristics of the minimum frequency point, and / or the reconstructed impedance equation. For the electrochemical model, based on the impedance spectrum of the model, the center and radius of the circle or arc in the Nyquist plot are calculated, and the initial model parameters are solved based on the center and radius data. S4. Based on the physical constraint optimization strategy for model construction parameters, the initial model parameters are used as the initial solution. The nonlinear least squares method is called to iteratively adjust the model parameters to obtain the optimal solution. In step S2, the adaptive model selection algorithm is called to match the impedance model for the circuit component model; The adaptive model selection algorithm includes: Calculate the initial phase angle ; The maximum and minimum points of the amplitude are detected, and the model is matched using the total number of extreme points N and the initial phase angle. ； As described above, impedance model A is the equivalent circuit model of a resistor, capacitor, and inductor connected in parallel; impedance model B is the equivalent circuit of an inductor and resistor connected in series and then connected in parallel with a capacitor; impedance model C is the equivalent circuit of a resistor, capacitor, and inductor connected in series; impedance model D is the equivalent circuit of an inductor connected in series with a parallel capacitor and resistor; and impedance model E is the equivalent circuit of an inductor, capacitor two, and resistor connected in series and then connected in parallel with capacitor one.

2. The nonlinear fitting method for the impedance equivalent model according to claim 1, characterized in that, In S3, for circuit component models (1) Solve for the initial model parameters based on the characteristics of the resonance point; including: Based on the maximum amplitude corresponding to the series resonant point and the minimum amplitude corresponding to the parallel resonant point, the resonant point is obtained by extreme value analysis of the sweep frequency data amplitude. The resonant point frequency ω_res is substituted into the imaginary part of the impedance function to establish the equation Im(Z)(ω_res)=0, and the initial model parameters are obtained by solving it. (2) Solve for the initial model parameters based on the characteristics of the minimum frequency point; including: For cases where a capacitor is connected in parallel or an inductor is connected in series in a composite circuit, a simplified model is obtained by treating the parallel capacitor as an open circuit and the series inductor as a short circuit under low-frequency conditions. The minimum frequency ω_min is substituted into the real and imaginary parts of the simplified model, and the real and imaginary part equations are established with the measured data respectively. The initial model parameters are obtained by solving the equations. (3) For models where the inductance cannot be directly determined from the characteristics of the resonant point and the minimum frequency point, the initial model parameters are solved by step-by-step calculation; including: Based on the determined initial model parameters, the impedance equation is reconstructed, and the explicit expression for the inductance parameter L is derived; the inductance estimate at each frequency point is solved by traversing the swept frequency data samples. ; The final initial value L_init = mean is obtained by using the arithmetic mean method. ).

3. The nonlinear fitting method for the impedance equivalent model according to claim 1, characterized in that, In S3, for the electrochemical model, a Nyquist plot of the model impedance spectrum is established, and the center and radius of the circles or arcs in the plot are calculated, including: For composite components connected in parallel with RC circuits: ； ， ； The coordinates of the center of the circle are (R / 2, 0), and the radius is R / 2. For composite components with R and CPE connected in parallel: ， ， ， The coordinates of the center and the radius are obtained as follows: , ;in, The equivalent capacitance of the load-transfer structure. These are the load-transfer structural characteristic values; R is the resistance value, and C is the capacitance value. For current / voltage frequency, This is the real part of the impedance. This represents the imaginary part of the impedance. The initial model parameters are solved based on the center and radius data, including: Initial resistance value: On the Nyquist plot, an interpolation algorithm is used to determine a value that intersects the horizontal axis of the real part as the initial value of RS. ; Initial inductance value: Solved using the imaginary part of the highest frequency point, it is equal to the imaginary part of the impedance at the highest frequency point divided by the corresponding angular frequency; Initial values ​​of parallel composite elements: Calculate the initial resistance value. The real part of the impedance at the maximum point peak_index; by Calculate the initial value of the capacitor. The frequency of the maximum point peak_index; For the R-CPE parallel circuit, based on the imaginary part and frequency of the maximum point of the irregular circular arc segment, as well as the center and radius of the circle, the initial values ​​of α and Q are further obtained. , Where peak_index is the maximum point of the negative imaginary part. The initial value of the resistance. ; This is the imaginary part of the impedance at the point of maximum value.

4. A nonlinear fitting system for an impedance equivalent model, characterized in that, include: The frequency sweeping hardware module is used to acquire measurement signals at each frequency point based on the configured frequency sweeping parameters; The data processing module is used to calculate the impedance values ​​at various frequencies and select an appropriate impedance model based on the device under test. According to the selected impedance model, it calls a multiphysics coupling prediction algorithm to predict the initial model parameters of the impedance model. Based on the physical constraint optimization strategy for model construction parameters, using the initial model parameters as the initial solution, it calls a nonlinear least squares method to iteratively adjust the model parameters to obtain the optimal solution. The multiphysics coupling prediction algorithm includes: for circuit component models, solving for the initial model parameters using a step-by-step derivation method based on resonant point characteristics, minimum frequency point characteristics, and / or reconstructed impedance equations; for electrochemical models, calculating the center and radius of the circles or arcs in the Nyquist plot based on the model's impedance spectrum, and solving for the initial model parameters based on the center and radius data. The display module allows for frequency sweep configuration via a UI interface and displays the fitting results and responses in the form of Nyquist plots or Bode plots. The data processing module calls an adaptive model selection algorithm to match the impedance model for the circuit component model. The adaptive model selection algorithm includes: Calculate the initial phase angle ; The maximum and minimum points of the amplitude are detected, and the model is matched using the total number of extreme points N and the initial phase angle. ； As described above, impedance model A is the equivalent circuit model of a resistor, capacitor, and inductor connected in parallel; impedance model B is the equivalent circuit of an inductor and resistor connected in series and then connected in parallel with a capacitor; impedance model C is the equivalent circuit of a resistor, capacitor, and inductor connected in series; impedance model D is the equivalent circuit of an inductor connected in series with a parallel capacitor and resistor; and impedance model E is the equivalent circuit of an inductor, capacitor two, and resistor connected in series and then connected in parallel with capacitor one.

5. The nonlinear fitting system for the impedance equivalent model according to claim 4, characterized in that, The data processing module is designed for circuit component models. (1) Solve for the initial model parameters based on the characteristics of the resonance point; including: Based on the maximum amplitude corresponding to the series resonant point and the minimum amplitude corresponding to the parallel resonant point, the resonant point is obtained by extreme value analysis of the sweep frequency data amplitude. The resonant point frequency ω_res is substituted into the imaginary part of the impedance function to establish the equation Im(Z)(ω_res)=0, and the initial model parameters are obtained by solving it. (2) Solve for the initial model parameters based on the characteristics of the minimum frequency point; including: For cases where a capacitor is connected in parallel or an inductor is connected in series in a composite circuit, a simplified model is obtained by treating the parallel capacitor as an open circuit and the series inductor as a short circuit under low-frequency conditions. The minimum frequency ω_min is substituted into the real and imaginary parts of the simplified model, and the real and imaginary part equations are established with the measured data respectively. The initial model parameters are obtained by solving the equations. (3) For models where the inductance cannot be directly determined from the characteristics of the resonant point and the minimum frequency point, the initial model parameters are solved by step-by-step calculation; including: Based on the determined initial model parameters, the impedance equation is reconstructed, and the explicit expression for the inductance parameter L is derived; the inductance estimate at each frequency point is solved by traversing the swept frequency data samples. ; The final initial value L_init = mean is obtained by using the arithmetic mean method. ).

6. The nonlinear fitting system for the impedance equivalent model according to claim 4, characterized in that, The data processing module, for the electrochemical model, establishes the Nyquist plot of the model impedance spectrum and calculates the center and radius of the circles or arcs in the plot, including: For composite components connected in parallel with RC circuits: ； ， ； The coordinates of the center of the circle are (R / 2, 0), and the radius is R / 2. For composite components with R and CPE connected in parallel: ， ， ， The coordinates of the center and the radius are obtained as follows: , ;in, The equivalent capacitance of the load-transfer structure. These are the load-transfer structural characteristic values; R is the resistance value, and C is the capacitance value. For current / voltage frequency, This is the real part of the impedance. This represents the imaginary part of the impedance. The initial model parameters are then solved based on the center and radius data, including: Initial resistance value: On the Nyquist plot, an interpolation algorithm is used to determine a value that intersects the horizontal axis of the real part as the initial value of RS. ; Initial inductance value: Solved using the imaginary part of the highest frequency point, it is equal to the imaginary part of the impedance at the highest frequency point divided by the corresponding angular frequency; Initial values ​​of parallel composite elements: Calculate the initial resistance value. The real part of the impedance at the maximum point peak_index; by Calculate the initial value of the capacitor. The frequency of the maximum point peak_index; For the R-CPE parallel circuit, based on the imaginary part and frequency of the maximum point of the irregular circular arc segment, as well as the center and radius of the circle, the values ​​of α and Q are further obtained. , Where peak_index is the maximum point of the negative imaginary part. The initial value of the resistance. ; This is the imaginary part of the impedance at the point of maximum value.

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