An interface reaction kinetics parameter analysis method based on electrochemical impedance spectroscopy

Abstract

The application discloses an interface reaction kinetics parameter analysis method based on electrochemical impedance spectroscopy, and relates to the technical fields of electrochemical testing and data processing, and comprises the following steps: applying an alternating current signal with gradually increased amplitude to a to-be-tested interface, and extracting a fundamental frequency impedance vector and a harmonic impedance vector; performing anti-convolution calculation on the fundamental frequency impedance vector and the harmonic impedance vector respectively by using preset standard penalty weights, obtaining a fundamental frequency relaxation vector and a harmonic relaxation vector, and dividing the amplitudes of the two vectors by a node to obtain a response ratio; identifying an aliasing peak interval with an information entropy greater than an entropy value determination threshold in the fundamental frequency relaxation vector, wherein the aliasing peak interval contains an aliasing peak, and the aliasing peak only overlaps with an extreme value point of the response ratio; and the method has the beneficial effect that the evaluation error caused by improper selection of a human prior model is reduced, and the reliability of interface kinetics parameter analysis is ensured.

Description

Technical Field

[0001] This invention relates to the field of electrochemical testing and data processing technology, and in particular to a method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy. Background Technology

[0002] Electrochemical impedance spectroscopy (EIS), as a non-destructive detection method, is widely used to study the kinetics of complex electrochemical interfaces such as energy storage devices, electrocatalytic reactions, and metal corrosion. In existing analytical systems, relaxation time distribution algorithms, because they do not require a pre-defined equivalent circuit model, can transform frequency-domain complex impedance data into a characteristic distribution in the relaxation time domain through deconvolution, thereby achieving a refined characterization of electrochemical polarization processes. Accurate extraction of characteristic parameters of different physicochemical processes at the interface is of significant guiding importance for evaluating the electrode process kinetics of electrochemical systems and optimizing material properties.

[0003] However, existing analytical methods based on linear deconvolution impedance spectroscopy suffer from significant resolution limitations in practical applications. When different polarization response mechanisms within a system are extremely similar on the physical timescale, existing analytical algorithms, relying solely on linear impedance data in a single frequency dimension, struggle to distinguish closely adjacent processes with similar integral kernels at the underlying mathematical logic level. This traditional mathematical fitting approach, combined with a globally unified smoothing constraint mechanism, cannot effectively overcome the physical resolution limit. Consequently, in the relaxation time distribution spectrum, characteristic peaks that should be independent often merge into broadened, overlapping peaks that are difficult to separate. Due to the lack of deep coupling constraints between the system's intrinsic physical properties and the mathematical inversion process, existing linear analytical logic often extracts severely distorted kinetic parameters when dealing with complex overlapping responses, leading to the failure of assessing the interface kinetic state. Therefore, it is necessary to design an analytical method for interface reaction kinetic parameters based on electrochemical impedance spectroscopy to address these issues. Summary of the Invention

[0004] In view of the above-mentioned prior art, this application is hereby filed. Embodiments of this application provide a method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy, which reduces evaluation errors caused by improper selection of prior models and ensures the reliability of interfacial kinetic parameter analysis.

[0005] According to one aspect of this application, a method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy is provided, comprising:

[0006] An AC signal with progressively increasing amplitude is applied to the interface under test, and the fundamental frequency impedance vector and harmonic impedance vector are extracted.

[0007] The fundamental frequency impedance vector and the harmonic impedance vector are deconvolved using preset standard penalty weights to obtain the fundamental frequency relaxation vector and the harmonic relaxation vector. The amplitudes of the two are then divided by the nodes to obtain the response ratio.

[0008] Identify the aliasing peak intervals in the fundamental frequency relaxation vector where the information entropy is greater than the entropy threshold, wherein the aliasing peak interval contains aliasing peaks, and the aliasing peak interval is confirmed as a deeply coupled interval only when the peak position of the aliasing peak overlaps with the extreme point of the response ratio.

[0009] A heterogeneous weight matrix is ​​generated based on the response ratio within the deep coupling interval: a penalty weight less than the standard penalty weight is assigned to nodes where the response ratio is greater than a preset ratio judgment value, and a penalty weight greater than or equal to the standard penalty weight is assigned to nodes that approach zero.

[0010] The heterogeneous weight matrix is ​​used to replace the standard penalty weight. Based on the replaced penalty weight, the fundamental frequency impedance vector is iteratively calculated to generate an updated fundamental frequency relaxation vector until the aliasing peak is separated into independent characteristic peaks, and the decoupled charge transfer resistance and double-layer capacitance are output.

[0011] According to another aspect of this application, an electronic device is provided, including a memory and a processor, the memory being used to store computer-executable instructions, and the processor being used to execute the computer-executable instructions, which, when executed by the processor, implement the steps of the method described above.

[0012] According to another aspect of this application, a computer storage medium is provided that stores computer-executable instructions thereon, which, when executed by a processor, implement the steps of the method described above.

[0013] Compared with the prior art, the method of analyzing interface reaction kinetic parameters based on electrochemical impedance spectroscopy according to the embodiments of this application utilizes the physical intrinsic differences between the nonlinear response characteristics of the charge transfer process and the linear response bias of the double-layer capacitance to obtain the response ratio of the fundamental frequency and harmonics, providing a reliable physical characteristic basis for separating overlapping physical processes. By using the cross-judgment condition of the overlap between the peak position of the aliasing peak and the extreme point of the response ratio, the interference caused by background noise or simple mathematical fitting error is eliminated.

[0014] At nodes with high response ratios (reflecting strong Faraday charge transfer characteristics), smaller penalty weights are assigned to allow for sharpening of the characteristic peaks; at nodes with response ratios approaching zero (reflecting pure capacitance characteristics), penalty weights are maintained or increased to keep the curve smooth. This local spatial mathematical intervention based on the asymmetry of physical response guides the iterative calculation to converge in a direction consistent with physical reality. The entire iterative calculation process does not require a pre-set equivalent circuit model. Driven by the local heterogeneous weight matrix, the originally aliased spectrum is separated into independent characteristic peaks. The final output charge transfer resistance and double-layer capacitance parameters are directly calculated based on the physical and mathematical characteristics of the data itself, reducing the evaluation error caused by improper selection of prior models and ensuring the reliability of the interface dynamics parameter analysis. Attached Figure Description

[0015] The above and other objects, features, and advantages of this application will become more apparent from the more detailed description of the embodiments of this application in conjunction with the accompanying drawings. The drawings are provided to further illustrate the embodiments of this application and form part of the specification. They are used together with the embodiments of this application to explain this application and do not constitute a limitation thereof. In the drawings, the same reference numerals generally represent the same components or steps.

[0016] Figure 1 This is a schematic diagram of the overall process of the method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy according to the present invention.

[0017] Figure 2 This is a schematic diagram of the logical framework of the interfacial reaction kinetic parameter analysis method based on electrochemical impedance spectroscopy of the present invention.

[0018] Figure 3 This is an extended schematic diagram of the method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy according to the present invention. Detailed Implementation

[0019] Hereinafter, exemplary embodiments according to this application will be described in detail with reference to the accompanying drawings. Obviously, the described embodiments are merely some embodiments of this application, and not all embodiments of this application. It should be understood that this application is not limited to the exemplary embodiments described herein.

[0020] Example 1:

[0021] In existing technologies, electrochemical impedance spectroscopy (EIS) and relaxation time distribution (DRT) algorithms are widely used for the refined characterization of electrochemical interface polarization processes. However, they still have significant resolution limitations when dealing with complex systems with similar time constants. When the charge transfer process and the double-layer charging and discharging process are very close in physical timescale, existing analytical algorithms, which rely solely on linear impedance data in a single frequency dimension for deconvolution, struggle to distinguish closely adjacent processes with similar integral kernels at the underlying mathematical logic level. This purely linear mathematical fitting, combined with a globally unified smoothing constraint mechanism, cannot overcome the physical resolution limit. Inevitably, this leads to the target parameters merging into inseparable broadened aliased peaks in the relaxation time distribution spectrum, resulting in severe distortion of the analysis of interface dynamic parameters such as charge transfer resistance and double-layer capacitance.

[0022] To address the aforementioned issues, the core reason why traditional analytical methods cannot break through the resolution limit lies in the lack of effective constraints on the mathematical inversion process by the underlying physical characteristics of purely linear deconstruction. The charge transfer process of the system follows nonlinear dynamics and is prone to generating high-order harmonics under perturbation, while the double-layer charging process is highly linear. The two have intrinsic differences in physical response. This invention proposes a technical route that combines multi-order impedance response extraction, nonlinear response ratio construction, and dynamic reconstruction of the penalty operator. Specifically, by applying a stepped AC signal, the fundamental frequency and harmonic impedance vectors are extracted and divided to obtain the response ratio. The information entropy and the extreme value coincidence condition are used to accurately lock the deeply coupled aliasing peak interval. Then, a heterogeneous weight matrix is ​​generated based on the response ratio distribution in this interval. The standard penalty weight in the deconvolution calculation is locally adaptively replaced and the closed-loop iteration is driven. This scheme effectively breaks the separation barrier of aliasing peaks by transforming the asymmetry of the electrochemical nonlinear physical response into the asymmetric penalty intervention in the underlying mathematical space. It achieves accurate decoupling of similar time constants and greatly improves the objectivity, accuracy, and reliability of the evaluation of complex interface reaction kinetics.

[0023] Reference Figures 1-3 As an embodiment of the present invention, a method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy is provided, comprising: S1-S5.

[0024] For ease of understanding, the positive electrode / electrolyte interface of a commercially available pouch lithium-ion battery will be used as the subject of this explanation. Assume this battery uses... (i.e., NCM523) positive electrode material paired with a graphite negative electrode, and the electrolyte is... of Soluble in a mixed solvent of ethylene carbonate (EC) and dimethyl carbonate (DMC) (volume ratio 1:1), with a rated capacity of 2.5 Ah and a rated voltage of 3.7 V; during testing, the battery's state of charge (SOC) remained stable at 50%, and the ambient temperature was constant at 25°C; the frequency range covered by the test was 10 MHz to 100 kHz, and the relaxation time discretization coverage range was [missing information]. Within the discretized relaxation time coverage range, 128 relaxation time nodes are set at equal intervals. In this system, the solid electrolyte interphase (SEI / CEI) membrane resistance process on the surface of active particles and the charge transfer process and double-layer charge-discharge process at the cathode / electrolyte interface are similar in time constant, and often merge into broadened mixed peaks that are difficult to distinguish in the traditional relaxation time distribution spectrum.

[0025] Figure 1 and Figure 2 The illustration shows a method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy according to an embodiment of this application, specifically including:

[0026] like Figure 1 As shown, in step S1, an AC signal with progressively increasing amplitude is applied to the interface under test, and the fundamental frequency impedance vector and harmonic impedance vector are extracted.

[0027] In this step, the amplitude of the AC signal is applied in a stepped manner to simultaneously excite the linear and nonlinear responses at the interface. Specifically, the amplitude of the AC excitation is set to several levels (e.g., 3-5 levels) in ascending order. Each level performs a complete frequency scan within the covered frequency scanning range (e.g., 10mHz-100kHz), and the current and voltage time-domain responses at each frequency are recorded respectively.

[0028] The fundamental frequency impedance vector is the vector composed of the steady-state impedance response at the corresponding excitation angular frequency for each excitation frequency. The harmonic impedance vector is the vector composed of the impedance response corresponding to the second or higher harmonic frequency components (corresponding to integer multiples of the excitation frequency) extracted after performing a discrete Fourier transform on the time-domain response. Specifically, the extraction method can be to use a fast Fourier transform (FFT) to convert the time-domain response of the current and voltage to the frequency domain, read the amplitude and phase of the fundamental frequency position and the harmonic position respectively, and divide them to obtain the corresponding impedance component.

[0029] It should be noted that the selection of excitation amplitude must take into account both the ability to induce a nonlinear response and the requirement to keep the system in a small disturbance state. Typically, the excitation amplitude does not exceed the order of magnitude of thermal voltage kT / e (approximately 25.7mV@25℃). The number and value of the amplitude levels can be adjusted according to the nonlinearity of the interface under test. Too few amplitude levels will be detrimental to the stable extraction of harmonic components, while too many amplitude levels may introduce additional distortion beyond the small signal assumption.

[0030] Using the aforementioned lithium-ion battery example, in actual testing, the excitation amplitude levels can be set sequentially to 5mV, 10mV, and 20mV. Each level is scanned within the range of 10mHz-100kHz at a density of 7 frequency points per decade, thereby obtaining the fundamental frequency impedance vector and the second harmonic impedance vector at the corresponding level. Through the above scheme, the extraction of the fundamental frequency impedance vector and the harmonic impedance vector respectively constitutes the data basis for separating the linear response and nonlinear response in subsequent steps.

[0031] return Figure 1 In step S2, the fundamental frequency impedance vector and harmonic impedance vector are deconvolved using preset standard penalty weights to obtain the fundamental frequency relaxation vector and harmonic relaxation vector, as shown in the following formula:

[0032] ;

[0033] ;

[0034] , and , respectively, are the fundamental frequency relaxation vector and the harmonic relaxation vector; x is the relaxation variable to be solved in the deconvolution process; and A represents the discretized integral kernel matrix, used to map the relaxation domain to the impedance domain. This represents the operator for finding the independent variable x that minimizes the objective function. This is the fundamental frequency impedance vector extracted from actual measurements. This is the measured harmonic impedance vector. The standard penalty weights are used to adjust the balance between fidelity and smoothness, and L represents the difference operator matrix, used to constrain the smoothness of the relaxation vector. The operator that represents the square of the second norm of a vector;

[0035] The essence of the above deconvolution calculation is to solve an ill-posed linear inverse problem under the Tikhonov regularization framework: the first term represents the fit fidelity of the solved relaxation vector to the measured impedance vector after transformation by the integral kernel matrix, and the second term represents the smoothing constraint of the relaxation vector itself between adjacent nodes. The two are weighted and adjusted by standard penalty weights.

[0036] Specifically, the integral operator matrix is ​​constructed based on the superposition assumption of Debye single relaxation processes, and its q-th column element corresponds to the relaxation time node. The impedance contribution coefficient of the unit relaxation response at each excitation frequency (typically in the form of 1 / (1+jω)). (or the corresponding real / imaginary components), the function of the entire matrix is ​​to map the weighted superposition of the relaxation intensity of each node in the relaxation time domain to the impedance response in the frequency domain. The difference operator matrix is ​​taken as a second-order discrete difference form, and its function is to perform second-order difference operations on the relaxation vector elements at adjacent relaxation time nodes, so that the relaxation vector can suppress non-physical violent oscillations between adjacent nodes during the solution process.

[0037] It should be noted that the numerical magnitude of the standard penalty weights typically falls within the range of... The specific values ​​can be determined using the L-curve method—plotting the relationship between the norm of the fidelity term and the norm of the smoothing term on logarithmic coordinates for different penalty weight values, and taking the value corresponding to the maximum curvature of the curve as the standard penalty weight; or using the generalized cross-validation (GCV) method, minimizing the prediction error through leave-one-out cross-validation; for test systems with established historical databases, the mean or median of the optimal values ​​obtained in previous tests can be used as the preset standard penalty weight.

[0038] Divide the two amplitudes by the node to obtain the response ratio, as shown in the following formula:

[0039] ;

[0040] in, Represents a node The response ratio at the specified point, where k is the discrete index subscript of the relaxation time node. This represents the k-th logarithmic relaxation time point. The harmonic relaxation vector at the node The value at that location, The fundamental frequency relaxation vector at the node The value at that location, Indicates taking the absolute value;

[0041] The response ratio reflects the relative strength of the harmonic response to the fundamental frequency response at each relaxation time point. At the physical level, the charge transfer process follows an exponential nonlinear relationship described by the Butler-Volmer equation, which easily generates second and higher harmonic components of a certain strength under AC disturbances. In contrast, the double-layer charging and discharging process exhibits a highly linear response characteristic under small disturbances, producing almost no harmonic contribution. Therefore, the response ratio is relatively large near the relaxation time point dominated by the charge transfer process, and approaches zero near the relaxation time point dominated by the double-layer charging and discharging process. The distribution curve of the response ratio on the relaxation time axis provides an auxiliary criterion independent of linear impedance data for distinguishing between two types of processes with different intrinsic physical responses but similar time constants.

[0042] Following the previous example, the fundamental frequency impedance vector and the second harmonic impedance vector obtained by the battery under three excitation amplitudes are deconvolved and calculated. The standard penalty weight determined by the L-curve method is approximately... The fundamental frequency relaxation vector obtained by deconvolution takes approximately [time missing] in the logarithmic relaxation time. At this point, a mixed peak with a relatively wide half-width is observed. The harmonic relaxation vector within the same interval also exhibits a response, but its peak position is slightly biased towards the shorter side of the logarithmic time axis. The response ratio obtained by dividing the two by the nodes forms a local extremum near this location, with the ratio at the extremum being approximately... The ratio approaches zero within the range far from the extreme value. Through the above scheme, the fundamental frequency relaxation vector, harmonic relaxation vector and response ratio together constitute the basic data object for subsequent coupling identification and heterogeneous weight construction.

[0043] return Figure 1 In step S3, the aliasing peak intervals in the fundamental frequency relaxation vector where the information entropy is greater than the entropy value determination threshold are identified, wherein the aliasing peak intervals contain aliasing peaks.

[0044] In this step, the information entropy is calculated using the Shannon entropy form, obtained by normalizing the amplitude of the fundamental frequency relaxation vector. This involves dividing the relaxation intensity at each node by the sum of the vectors to obtain a normalized probability distribution, and then calculating the information entropy for each node based on this probability distribution. The resulting value is the information entropy. The information entropy corresponding to a single narrow peak is close to zero, while the information entropy corresponding to a broadened aliasing peak is close to... (where N is the number of nodes contained in the interval), thus the magnitude of the information entropy can reflect the degree of widening of the relaxation distribution within the interval;

[0045] It should be noted that the specific value of the entropy threshold can be determined based on historical test data statistics: calculate the normalized entropy value for the interval containing known unimodal data and the interval containing typical mixed peak data respectively, and take the distinguishing point of the entropy distribution of the two types of samples as the entropy threshold; or it can be determined empirically. (where N is the number of nodes contained in the interval,) for (Empirical coefficients between them);

[0046] The aliasing peak interval is confirmed as a deeply coupled interval only when the peak position of the aliasing peak overlaps with the extreme point of the response ratio, including:

[0047] The zero points of the first derivative of the fundamental frequency relaxation vector and the zero points of the first derivative of the response ratio within the aliasing peak interval are obtained respectively, thus obtaining the fundamental frequency peak node and the ratio extreme value node;

[0048] The zero point of the first derivative corresponds to the extreme value position of the vector curve. The zero point of the first derivative of the fundamental frequency relaxation vector is the node where the peak of the aliasing peak is located. The zero point of the first derivative of the response ratio is the node where the extreme value of the response ratio is located. In discrete numerical processing, the first derivative can be calculated by forward difference, backward difference or central difference. The position where the difference sign of adjacent nodes changes from positive to negative is taken as the discrete approximation of the zero point.

[0049] Extracting the curvature features of the response ratio in the neighborhood of the fundamental frequency peak node, including:

[0050] Extract the adjacent local minimum nodes of the response ratio on both sides of the fundamental frequency peak node, and define the continuous interval between the two local minimum nodes as the neighborhood;

[0051] Specifically, the identification method for local minima is as follows: the response ratio sequence is differentially analyzed node by node. If the response ratio at a certain node is simultaneously less than the response ratio of the preceding node and the response ratio of the following node, then the node is defined as a local minima. "Adjacent" means that starting from the location of the fundamental frequency peak node, the search proceeds node by node along the short-time direction and the long-time direction of the relaxation time axis. The first node found that satisfies the aforementioned conditions is the adjacent local minima. Furthermore, the discrete index subscript of the adjacent local minima on the short-time direction side is used as the starting point of the neighborhood, and the discrete index subscript of the adjacent local minima on the long-time direction side is used as the ending point of the neighborhood. The set of all consecutive relaxation time nodes between the two index subscripts (including the two endpoints) is the neighborhood.

[0052] Calculate the second-order difference value of the response ratio at the fundamental frequency peak node, and extract the absolute value of the second-order difference value as the absolute sharpness. The specific formula is as follows:

[0053] ;

[0054] in, denoted as the absolute sharpness (dimensionless) at the fundamental frequency peak node, and p is the discrete index subscript corresponding to the fundamental frequency peak node. This represents the logarithmic relaxation time node corresponding to the fundamental frequency peak node. Represents a node Response ratio at the location, and Each is the next node and the previous node Response ratio, Indicates taking the absolute value;

[0055] Absolute sharpness reflects the sharpness of the response ratio curve at the peak position—the larger the value, the steeper the change in the response ratio near that node; the local evolution energy index reflects the overall energy level of the response ratio in the neighborhood of the peak position—the larger the value, the higher the cumulative amplitude of the response ratio in that neighborhood. The ratio of the two, as a curvature feature, can eliminate artificially high sharpness caused by an overly high overall response ratio level in the neighborhood (i.e., the sharpness is normalized), thus making the curvature feature comparable across intervals and samples. Alternatively, the curvature feature can also be extracted by using the second derivative extremum after kernel density estimation, the reciprocal of the peak width after Gaussian peak fitting, or the modulus maxima of wavelet transform coefficients, etc.

[0056] The response ratio within the neighborhood is numerically integrated to generate a local evolution energy index, as shown in the following formula:

[0057] ;

[0058] in, Let be the local evolution energy index (dimensionless), where a and b are the discrete index subscripts corresponding to the local minimum nodes on the left and right sides of the neighborhood, respectively, and k is the discrete index subscript of the relaxation time node. Represents a node Response ratio at the location, Indicates the step size between adjacent logarithmic relaxation time points ( ;

[0059] Calculate the ratio of the absolute sharpness index to the local evolution energy index, and use this ratio as the curvature feature. ,Right now ;

[0060] Calculate the Euclidean distance between the fundamental frequency peak node and the ratio extreme value node, i.e.:

[0061] ;

[0062] in, denoted as the Euclidean distance between the fundamental frequency peak node and the ratio extreme value node. This represents the logarithmic relaxation time node corresponding to the fundamental frequency peak node. This represents the logarithmic relaxation time point corresponding to the extreme value node of the ratio;

[0063] The aliasing peak interval is confirmed as a deep coupling interval only when the curvature feature is greater than the preset curvature threshold and the Euclidean distance is less than the preset deviation threshold.

[0064] The physical meaning of the dual criteria is as follows: if the curvature feature is greater than the preset curvature threshold, it indicates that the response ratio has formed a sharp extreme value feature in the neighborhood of the fundamental frequency peak node, reflecting the presence of a nonlinear response peak contributed by the charge transfer process at this position, thus eliminating false extreme values ​​corresponding to the slight fluctuations in the response ratio caused by random noise in the harmonic extraction process; if the Euclidean distance is less than the preset deviation threshold, it indicates that the peak position of the aliasing peak in the fundamental frequency relaxation vector and the extreme point of the response ratio point to the same position on the logarithmic relaxation time axis, that is, the physical processes corresponding to the two are consistent in time constant. Only when both conditions are met can it be determined that the formation of the aliasing peak includes the superposition of the charge transfer process and the double-layer charging and discharging process under similar time constants, thus confirming the interval as a deeply coupled interval that needs to be processed by subsequent heterogeneous weighting; if either condition is not met, the identified aliasing peak may be caused by data fluctuations, broadening of the shape of a single process peak, or interference from other physical processes, and is not included in the construction range of the subsequent heterogeneous weighting matrix.

[0065] It should be noted that the specific value of the curvature threshold can be determined based on historical test data statistics: calculate the distribution of curvature features for samples with aliasing peaks known to be in deeply coupled intervals and samples known to be in shallowly coupled or uncoupled intervals, and take the quantile boundary point of the two types of sample distributions as the entropy value judgment threshold; or take the average value of the curvature feature distribution in historical data plus one standard deviation as the threshold; the deviation threshold can be set according to the deviation step size Δ of the relaxation time node, for example, taking it to be on the order of 2Δ-3Δ, so as to control the tolerance of Euclidean distance judgment within the range of several adjacent nodes.

[0066] To address the difficulty in distinguishing closely adjacent time constant processes in the fundamental frequency relaxation vector obtained from linear deconvolution, this step uses a dual condition of "information entropy criterion + peak position extreme value coincidence criterion" to confirm the deep coupling interval. Relying solely on the information entropy criterion can easily misjudge background noise or data fluctuations as aliasing peaks; relying solely on the response ratio extreme point criterion may be affected by noise interference during harmonic extraction. Combining the two utilizes both the irregular shape of the fundamental frequency relaxation vector itself and the nonlinear physical information carried by the harmonic response. This allows for the location of aliasing peak intervals after eliminating background noise and simple mathematical fitting errors. While locating the deep coupling interval, it also eliminates false aliasing intervals caused by simple data fluctuations or noise, providing reliable interval location for the subsequent construction of the heterogeneous weight matrix.

[0067] Alternatively, the determination of the overlapping peak interval can also be achieved by using the half-peak width threshold method of the fundamental frequency relaxation vector, peak shape symmetry test, or Gaussian fitting residual test; the comparison of the peak position and the extreme point position can also be replaced by the correlation coefficient method or mutual information criterion. Considering both noise robustness and physical interpretability, this embodiment selects the above-mentioned combination criterion of information entropy and extreme value coincidence.

[0068] Following the previous example, after calculating the information entropy of the fundamental frequency relaxation vector, it is identified that the frequency is located at approximately the logarithmic relaxation time. There exists a region in the vicinity where the information entropy exceeds the entropy threshold; further, by extracting the zero point of the first derivative of the fundamental frequency relaxation vector within this region, the relaxation time corresponding to the fundamental frequency peak node is approximately... The relaxation time corresponding to the extreme node of the ratio where the first derivative of the response ratio is zero is approximately... The Euclidean distance between the two in logarithmic coordinates is approximately 0.03, which is less than the deviation threshold set in this example; the curvature feature calculated in the neighborhood of the fundamental frequency peak node is approximately 15, which is greater than the curvature threshold set in this example; both conditions are met simultaneously, thus confirming that this interval is a deeply coupled interval.

[0069] return Figure 1 In step S4, a heterogeneous weight matrix is ​​generated based on the response ratio within the deep coupling interval: nodes with a response ratio greater than a preset ratio judgment value are assigned a penalty weight less than the standard penalty weight, and nodes approaching zero are assigned a penalty weight greater than or equal to the standard penalty weight. Specifically, this includes:

[0070] First, it should be noted that in this step, the specific value of the preset ratio judgment value can be set based on the maximum value of the response ratio within the deep coupling range, for example, between 30% and 50% of the maximum response ratio; the criterion for "approaching zero" can be set as the response ratio not exceeding 5% to 10% of its maximum value;

[0071] Obtain the statistical mean and statistical variance of the response ratio within the deep coupling interval, including:

[0072] Node-wise differencing is performed on the response ratio within the deeply coupled region to extract the local gradient feature values ​​of each node. ,Right now: ,in, For the previous node Response ratio at the location;

[0073] Construct a continuous confidence coefficient that is negatively correlated with the absolute value of the local gradient eigenvalues, i.e. ,in, For nodes The continuity confidence coefficient at the point. To adjust the constant mapping coefficients for gradient sensitivity, The absolute value of the local gradient eigenvalue;

[0074] The continuity confidence coefficient reflects the local smoothness of the response ratio curve at each node. In regions where the response ratio changes gently, the absolute value of the local gradient eigenvalue is small, and the corresponding continuity confidence coefficient is large, indicating that the response ratio in this region is highly reliable. In regions where the response ratio changes drastically, the absolute value of the local gradient eigenvalue is large, and the corresponding continuity confidence coefficient is small, indicating that this region may be affected by data fluctuations or noise. Therefore, introducing the continuity confidence coefficient as a weight when calculating the statistical mean and statistical variance can reduce the interference of fluctuating nodes on the overall statistics, making the subsequent penalty weight construction more closely match the true distribution characteristics of the response ratio within the deeply coupled interval. The constant mapping coefficient for adjusting the gradient sensitivity can be calibrated according to the magnitude of the response ratio. For example, the continuity confidence coefficient obtained after substituting the typical absolute value of the gradient of the response ratio in this interval should fall into a moderate range (e.g., 0.3-0.7).

[0075] Using the continuity confidence coefficient as the weight, a weighted average is calculated for the response ratios within the deep coupling interval to obtain the weighted center value, which is then determined as the statistical mean. The specific formula is as follows:

[0076] ;

[0077] in, Here, m and n represent the weighted center values ​​(i.e., the statistical mean), and m and n are the discrete index subscripts corresponding to the start and end nodes of the deeply coupled interval, respectively. For nodes The continuity confidence coefficient at the point. Represents a node Response ratio at the location;

[0078] Calculate the squared deviation between the response ratio of each node and the statistical mean. Use the continuity confidence coefficient to calculate the weighted average of the squared deviations to obtain the weighted dispersion, and then define the weighted dispersion as the statistical variance. The specific formula is as follows:

[0079] ;

[0080] in, This is the weighted dispersion (i.e., statistical variance). For nodes The continuity confidence coefficient at the point. Represents a node Response ratio at the location;

[0081] The statistical mean is used as the numerical switching center for the penalty weight, and the numerical values ​​with positive correlation of statistical variance are used as the switching steepness factor. The penalty weight distribution curve is constructed based on the numerical switching center and the switching steepness factor.

[0082] The specific form of the penalty weight distribution curve can be constructed using a sigmoid function: with the response ratio as the independent variable and the penalty weight as the dependent variable, the statistical mean is used as the numerical switching center of the sigmoid curve (i.e., the penalty weight distribution curve) (i.e., when the response ratio equals the statistical mean, the corresponding penalty weight is exactly the median of its upper and lower limits), and a value positively correlated with the statistical variance is used as the switching steepness factor of the sigmoid curve (the larger the statistical variance, the wider the switching interval and the smoother the curve transition; the smaller the statistical variance, the narrower the switching interval and the steeper the curve transition). The upper limit of the sigmoid curve is taken as a value greater than or equal to the standard penalty weight (used to assign a larger penalty weight when the response ratio approaches zero to keep the curve smooth), and the lower limit is taken as a value less than the standard penalty weight (used to assign a smaller penalty weight when the response ratio is large to allow the characteristic peak to sharpen).

[0083] Alternatively, the penalty weight distribution curve can also be constructed using an arctangent function, an error function, or piecewise linear interpolation, as long as it satisfies the basic characteristics of "smaller values ​​in regions with larger response ratios, larger values ​​in regions with response ratios approaching zero, and smooth transition".

[0084] By using the penalty weight distribution curve, the response ratio within the deep coupling interval is converted into the corresponding penalty weight node by node.

[0085] Specifically, for each relaxation time node in the deep coupling interval, the response ratio at that node is substituted as the independent variable into the function corresponding to the penalty weight distribution curve. The value of the dependent variable calculated by the function mapping is the penalty weight corresponding to that node. By traversing all nodes in the deep coupling interval in ascending order of the discrete index subscripts of the relaxation time nodes, a penalty weight subsequence corresponding one-to-one with the nodes in the deep coupling interval can be obtained.

[0086] Furthermore, the heterogeneous weight matrix is ​​assembled as follows: for relaxation time nodes located within the deep coupling interval, the penalty weight subsequence obtained by mapping the response ratio through the penalty weight distribution curve is used; for relaxation time nodes located outside the deep coupling interval, the standard penalty weight used in step S2 is retained without change; the penalty weights of nodes within the deep coupling interval and the standard penalty weights of nodes outside the deep coupling interval are arranged in ascending order of the discrete index subscripts of the relaxation time nodes to form a weight sequence equal to the number of all relaxation time nodes. The diagonal matrix constructed with this weight sequence as the diagonal element and the remaining elements as zero is the heterogeneous weight matrix.

[0087] To address the issue that the globally uniform smoothing constraint in traditional deconvolution calculations leads to the inability to separate aliasing peaks, this step maps the intrinsic differences in the nonlinear responses of the charge transfer process and the double-layer charging and discharging process to the mathematical solution space, forming a locally adaptive heterogeneous penalty constraint. Specifically, nodes with larger response ratios correspond to regions with significant Faraday charge transfer characteristics, and are given smaller penalty weights to allow the characteristic peaks in these regions to be sharpened during iterative solutions. Nodes with response ratios approaching zero correspond to regions with significant double-layer charging and discharging characteristics, and the penalty weights are maintained or increased to preserve the smoothness of the solution results in these regions.

[0088] Thus, the smooth constraint that was originally uniformly applied to the entire solution space is transformed into a heterogeneous constraint that intervenes locally based on the differences in physical response, so that the deconvolution process converges in a direction that is consistent with the physical reality. Alternatively, the local heterogeneity of the penalty weight can also be achieved by setting piecewise constants, local weighting based on wavelet coefficients, or mapping functions trained based on historical data. Taking into account the adaptive correspondence with the response ratio and the continuity of numerical calculation, this embodiment selects the following curve mapping method based on statistical features.

[0089] Following the previous example, the local gradient feature value is obtained by calculating the difference of the response ratio node by node within the previously confirmed deep coupling interval, and then a continuity confidence coefficient is constructed. After weighted statistics, the weighted mean of the response ratio is approximately 0.18 and the weighted variance is approximately 0.06. Using this weighted mean as the numerical switching center and the value positively correlated with the weighted variance as the switching steepness factor, an S-shaped curve is constructed. The upper limit of the S-shaped curve is set to 1.2 times the standard penalty weight and the lower limit is set to 0.2 times the standard penalty weight. Then, the response ratio of each node in the deep coupling interval is substituted into the curve node by node to obtain the corresponding heterogeneous penalty weight set, which is then assembled to form a heterogeneous weight matrix.

[0090] The heterogeneous weight matrix constructed based on the statistical characteristics of the response ratio within the interval can map the local differences in the physical response to the constraint terms in the deconvolution solution process in a continuous and stable form, thus avoiding the interference of weight mutations on the solution process.

[0091] return Figure 1 In step S5, the standard penalty weights are replaced by a heterogeneous weight matrix. Based on the replaced penalty weights, the fundamental frequency impedance vector is iteratively calculated to generate an updated fundamental frequency relaxation vector until the aliasing peaks are separated into independent characteristic peaks, and the decoupled charge transfer resistance and double-layer capacitance are output.

[0092] Specifically, the iterative computation process includes the following loop: First, replace the standard penalty weights, originally used as scalars, in the deconvolution objective function of step S2 with a heterogeneous weight matrix. This transforms the smoothing constraint term in the objective function from a L2 norm squared form after the difference operator matrix is ​​applied to the relaxation vector to be calculated, into a weighted L2 norm squared form after the difference operator matrix is ​​applied to the relaxation vector to be calculated and then multiplied by the diagonal elements of the heterogeneous weight matrix node by node. Second, solve the replaced objective function using methods such as nonnegative least squares, conjugate gradient, quasi-Newton method, or interior point method. The optimal solution obtained by numerical optimization method is the updated fundamental frequency relaxation vector in the current iteration. The third step is to determine whether the updated fundamental frequency relaxation vector has been separated into independent characteristic peaks in the original aliasing peak interval. If it has not been separated, the updated fundamental frequency relaxation vector replaces the original fundamental frequency relaxation vector and the response ratio calculation in step S2, the deep coupling interval determination in step S3, and the heterogeneous weight matrix construction in step S4 are re-executed. The newly constructed heterogeneous weight matrix is ​​then substituted into the deconvolution objective function to solve. This process is repeated until the separation condition or convergence condition is met.

[0093] The convergence condition for iterative operations can be determined by the following criterion: the relative change between the relaxation vectors obtained from two consecutive iterations is less than a preset convergence threshold (e.g., ...). (Order of magnitude); or if two independent zeros of the first derivative appear within the overlapping peak region and there is a local minimum of the first derivative with opposite directions between the two zeros, it means that the characteristic peak has been separated into two independent characteristic peaks;

[0094] After the characteristic peaks are separated, the charge transfer resistance is taken as the integral of the relaxation intensity of the corresponding characteristic peak along the logarithmic relaxation time (i.e., the area below the characteristic peak); the double-layer capacitance is taken as the quotient obtained by dividing the relaxation time of the corresponding characteristic peak in the double-layer charging and discharging process by the corresponding equivalent resistance; alternatively, the above iterative solution process can also be implemented by optimization algorithms such as the alternating direction multiplier method (ADMM) and the near-end gradient method; the criterion for characteristic peak separation can also be replaced by the half-peak width non-intersection criterion or the valley depth criterion.

[0095] To address the non-uniformity of the deconvolution solution process after introducing the heterogeneous weight matrix, this step employs an iterative replacement and solution method to generate the separated relaxation vector. Specifically, in the deconvolution objective function of step S2, the original standard penalty weights, which were used as scalars, are replaced with a set of heterogeneous penalty weights (i.e., the heterogeneous weight matrix) that corresponds one-to-one with the relaxation time nodes, transforming the smoothing constraint term into a weighted L2 norm with node weights. Solving this objective function yields the updated fundamental frequency relaxation vector. If the aliasing peaks in the updated fundamental frequency relaxation vector have not yet been separated into independent characteristic peaks, the response ratio is recalculated based on the new relaxation vector, the heterogeneous weight matrix is ​​updated, and the solution is applied again. This process is iterated until convergence, completing the separation of two processes with similar time constants that were originally in an aliasing state, and outputting the corresponding charge transfer resistance and double-layer capacitance quantization parameters.

[0096] Following the previous example, substituting the constructed heterogeneous weight matrix into the deconvolution objective function, after the first iteration, the broadened aliasing peaks within the deep coupling region exhibit a trend of bimodal separation. After 3-5 iterations, the two independent feature peaks separate, with the feature peak on the side with the shorter relaxation time (peak position approximately...) This process is identified as being dominated by the charging and discharging of the electric double layer, with a characteristic peak (peak position approximately) on the side of the longer relaxation time. This process is identified as charge transfer-dominated; integrating the characteristic peaks on the long relaxation time side with the logarithmic relaxation time yields a charge transfer resistance of approximately [value missing]. The magnitude; based on the relaxation intensity distribution of the characteristic peak on the short relaxation time side, the double-layer capacitance is calculated to be approximately... Magnitude.

[0097] The deep coupling interval determination and heterogeneous weight matrix constructed in steps S1 to S5 above are all based on the response ratio derived from the fundamental frequency impedance vector and harmonic impedance vector collected under a single test condition as a characterization of the physical response difference. However, in actual testing, external conditions such as ambient temperature, applied overpotential, and state of charge of the interface under test may drift or be disturbed. The response ratio under a single condition reflects a relatively limited intensity of physical response difference at the boundary of the deep coupling interval or in the transition region on both sides of the aliasing peak. For interfaces with weak nonlinear response, the response ratio under a single condition is insufficient to stably highlight the charge transfer characteristics from the double-layer charging and discharging characteristics. This application further introduces a modulation vector derived from external disturbance conditions to inject multi-dimensional physical response information such as instantaneous phase, amplitude drift, and frequency dependence under external disturbances into the fundamental frequency relaxation vector, so as to enhance the distinguishability of charge transfer characteristics in the deep coupling interval in subsequent response ratio calculation and heterogeneous weight matrix construction. Figure 3 As shown, after obtaining the fundamental frequency relaxation vector, the process also includes:

[0098] Acquire the instantaneous phase characteristics, amplitude drift characteristics, and frequency dependence characteristics of the interface under test under temperature change conditions or electrochemical excitation conditions;

[0099] In this step, temperature change conditions refer to applying a controllable perturbation to the ambient temperature of the interface to be tested, such as applying a temperature step of ±5℃ or a temperature ramp at a rate of 1℃ / min based on the base test temperature; electrochemical excitation conditions refer to applying a perturbation to the interface to be tested that controls the potential / current, such as applying a potential step of ±10mV, a small overpotential pulse, or a small current step.

[0100] Instantaneous phase characteristics refer to the impedance phase angle at each relaxation time node in the corresponding frequency band, reflecting the capacitive / resistive response bias of the relaxation process at that node; amplitude drift characteristics refer to the ratio of the change in impedance amplitude at the same relaxation time node before and after an external disturbance, reflecting the sensitivity of that node to the amplitude of external disturbances; frequency dependence characteristics refer to the slope of the impedance amplitude at the same relaxation time node as the excitation frequency changes, reflecting the frequency domain dispersion of the response at that node.

[0101] The instantaneous phase characteristics, amplitude drift characteristics, and frequency dependence characteristics are weighted and summed to generate the modulation vector. The specific formula is as follows:

[0102] ;

[0103] in, For nodes The modulation vector value at that location. Represents a node Instantaneous phase characteristics at the location, Represents a node Amplitude drift characteristics at that location Represents a node Frequency-dependent features at the location, , and These are the adjustment weight constants for instantaneous phase, amplitude drift, and frequency dependence characteristics, respectively.

[0104] It should be noted that the specific values ​​of the adjustment weight constants for instantaneous phase, amplitude drift, and frequency dependence features can be determined in the following way: First, normalize the three types of features to the same magnitude range (e.g., [0,1]); second, in the existing calibration samples, take the fusion effect of the weighted sum of the three types of features and the fundamental frequency relaxation vector (e.g., the separability of feature distribution within the deep coupling interval after fusion) as the target, and use grid search or least squares fitting to calibrate the three weight constants; commonly used values ​​satisfy that the sum of the three weights is 1, and the adjustment weight constant of the instantaneous phase feature is usually slightly higher than the adjustment weight constants of the other two, because the phase deviation has a more direct indicative effect on the capacitance / resistance characteristics;

[0105] The modulation vector and the fundamental frequency relaxation vector are fused node by node to identify highly coupled regions in the fused data where the gradient change rate is greater than the gradient determination threshold.

[0106] Node-by-node fusion can be achieved through node-by-node multiplication, addition, or weighted superposition. This embodiment uses node-by-node multiplication as a typical implementation method, which takes into account both the intensity distribution of the fundamental frequency relaxation vector and the characterization of the physical deviation of the modulation vector. The gradient change rate of the fused data can be characterized by the absolute value of the difference between adjacent nodes. The gradient determination threshold can be taken as the upper quantile (e.g., 75%-90% quantile) of the gradient change rate distribution of the fused data, so as to identify the node region with the most dramatic changes as the high coupling interval.

[0107] The signal characteristics within the high-coupling region are numerically amplified to generate an enhanced relaxation vector, including:

[0108] Extract the instantaneous phase features corresponding to each relaxation time node within the high coupling interval, and calculate the absolute phase deviation between the instantaneous phase features and the ideal capacitor phase reference value. The specific formula is as follows:

[0109] ;

[0110] in, Represents a node The absolute phase deviation at that point Represents a node Instantaneous phase characteristics at the location, This represents the ideal capacitor phase reference value;

[0111] Construct an asymmetric amplification factor that is positively correlated with the absolute phase deviation;

[0112] The construction principle of the asymmetric amplification factor is as follows: when the absolute phase deviation is small (biased towards capacitance characteristics), the amplification factor is close to 1 (almost no amplification); when the absolute phase deviation is large (biased towards resistance characteristics), the amplification factor is greater than 1. This achieves targeted enhancement of the resistance-dominant node while not interfering with the capacitance-dominant node. Specifically, it can take the form of linear positive correlation with the absolute phase deviation (e.g., 1 plus the product of the asymmetric amplification gain factor and the absolute phase deviation), exponential function, or piecewise linear function. The value of the asymmetric amplification gain factor can be constrained according to the principle that the relaxation strength after amplification needs to be maintained at a reasonable physical order of magnitude. Usually, the value is chosen so that the maximum amplification factor falls between 2 and 5 times.

[0113] The magnitude of the fundamental frequency relaxation vector within the high-coupling interval is multiplied node-by-node by the asymmetric amplification factor to generate the enhanced relaxation vector, as shown in the following formula:

[0114] ;

[0115] in, Represents a node The number of enhanced relaxation vectors at that point. The fundamental frequency relaxation vector at the node The value at that location, The asymmetric amplification gain coefficient is the product of the absolute phase deviation. Represents a node Absolute phase deviation at the location;

[0116] Update response ratio and heterogeneous weight matrix based on enhanced relaxation vector;

[0117] Specifically, updating the response ratio based on the enhanced relaxation vector means replacing the original fundamental frequency relaxation vector with the enhanced relaxation vector as the denominator vector in the response ratio calculation formula, keeping the harmonic relaxation vector unchanged as the numerator vector, and obtaining the response ratio values ​​at each relaxation time node by dividing by the nodes and taking the absolute value, thus forming an updated response ratio sequence. Updating the heterogeneous weight matrix based on the enhanced relaxation vector means re-executing the deep coupling interval determination process in step S3 with the updated response ratio, including re-acquiring the fundamental frequency peak node and the ratio extreme value node, recalculating the curvature feature and Euclidean distance, and determining the threshold, to obtain the updated deep coupling interval; then, re-executing the statistical mean, statistical variance calculation, and construction of the penalty weight distribution curve in step S4 within the updated deep coupling interval with the updated response ratio, converting the response ratio of each node in the deep coupling interval into the corresponding penalty weight, and combining it with the standard penalty weight of the nodes outside the deep coupling interval according to the aforementioned assembly method to form the updated heterogeneous weight matrix.

[0118] In actual testing, the electrochemical properties of the interface under test are affected by external disturbances such as temperature, overpotential, and state of charge, resulting in variations in the response of the same interface under different conditions. To improve the robustness of deep coupling region determination under complex external disturbances, a modulation vector derived from the external disturbance conditions is fused with the fundamental frequency relaxation vector. This injects the physical information reflected by the external disturbance into the solution process, aiding in the identification of weak response characteristics within the deep coupling region. The physical response information under external disturbance conditions is injected into the fundamental frequency relaxation vector in the form of a modulation vector, and asymmetric amplification is applied to nodes deviating from ideal capacitance characteristics within the high coupling region. The resulting enhanced relaxation vector more prominently characterizes charge transfer features within the deep coupling region.

[0119] Following the previous example, a temperature step of +5°C was applied to the basic 25°C test conditions. The instantaneous phase, amplitude drift, and frequency dependence characteristics at each relaxation time point before and after the disturbance were recorded. The three values ​​were then weighted and summed to obtain the modulation vector. After multiplying the modulation vector and the fundamental frequency relaxation vector node by node, the high coupling interval was identified as being located at approximately the logarithmic relaxation time. The fundamental frequency relaxation vector within the interval is multiplied node by node according to the asymmetric amplification coefficient corresponding to the phase deviation to obtain the enhanced relaxation vector. Based on the enhanced relaxation vector, the response ratio is recalculated and the heterogeneous weight matrix is ​​updated. After substituting it into the deconvolution iteration again, the separation clarity between the charge transfer characteristic peak and the double electric layer characteristic peak is improved compared with the unmodulated case.

[0120] The response ratio obtained in step S2 above is constructed under the assumption that the harmonic impedance vector extraction process is undisturbed and the deconvolution solution of the harmonic relaxation vector is stable. However, in actual testing and data processing, the amplitude of the harmonic components is usually 1 to 2 orders of magnitude smaller than that of the fundamental frequency component. Their extraction results are more susceptible to the influence of background noise in the data acquisition system, fast Fourier transform truncation errors, and the unwell-posedness of the deconvolution solution. This results in abnormal jumps or local spikes at several relaxation time points in the response ratio sequence. If such local disturbances are not identified and directly participate in the deep coupling interval determination and heterogeneity analysis... The construction of the weight matrix may lead to a shift in the determination results of the deeply coupled interval, or cause the statistical mean and statistical variance in the penalty weight distribution curve to deviate from the true distribution characteristics of the response ratio within the interval due to the influence of outlier nodes. This application further decomposes the response ratio into multiple scales, identifies stable evolution segments and locally perturbed segments in the response ratio sequence through structured feature indicators at different scales, and reconstructs the response ratios located in the perturbed segments by numerical scaling. This allows for targeted correction of local perturbations in the response ratio before constructing the heterogeneous weight matrix. After obtaining the response ratio, the application also includes:

[0121] The response ratio is decomposed into multiple scales to extract local fluctuation features, phase coupling features, and energy distribution features at different scales.

[0122] Multi-scale decomposition can be achieved using discrete wavelet transform (DWT, such as decomposition to 3-5 scales based on the Daubechies wavelet basis), empirical mode decomposition (EMD), or variational mode decomposition (VMD). This embodiment uses discrete wavelet transform as a typical implementation method. The detail coefficients at different scales after decomposition correspond to the local variation information of the response ratio at different time scales.

[0123] Specifically, the local fluctuation feature is taken as the local mean of the absolute value of the detail coefficients at each scale within the neighborhood of the corresponding node, reflecting the short-term fluctuation intensity of the response ratio at that scale; the phase coupling feature is taken as the proportion of the same sign of the detail coefficients of the response ratio and the detail coefficients of the fundamental frequency relaxation vector at each scale, reflecting the phase synchronization degree of the response ratio and the fundamental frequency relaxation vector at that scale; the energy distribution feature is taken as the proportion of the cumulative energy of the square of the detail coefficients at each scale within the neighborhood of the corresponding node to the total energy of that node, reflecting the energy proportion of the response ratio at that scale.

[0124] Weighted mapping calculations are performed on local wave characteristics, phase coupling characteristics, and energy distribution characteristics to construct structural stability indices and disturbance characteristic indices for the response ratio, respectively. The specific formulas are as follows:

[0125] ;

[0126] ;

[0127] in, For nodes Structural stability indicators at the location. For nodes Disturbance characteristics at the location. For nodes The value of the local fluctuation characteristics at that location, For nodes The value of the phase coupling characteristic at that point, For nodes The value of the energy distribution characteristics at that location, , and All are stability feature mapping coefficients. , and All are perturbation characteristic mapping coefficients;

[0128] It should be noted that the stability characteristic mapping coefficients , and Mapping coefficients with perturbation characteristics , and The determination of the coefficients can be carried out as follows: First, normalize the local fluctuation characteristics, phase coupling characteristics, and energy distribution characteristics respectively; second, calculate the typical values ​​of the three types of characteristics based on the labeled stable and perturbation samples respectively; third, determine the sign and magnitude of the coefficients according to the contribution direction of the three types of characteristics to stability / perturbation (for example, higher phase coupling characteristics, higher energy distribution characteristics, and lower local fluctuation characteristics usually correspond to the structural stability interval; conversely, they correspond to the perturbation interval); finally, normalization constraints are applied with the sum of stability feature mapping coefficients being 1 and the sum of perturbation feature mapping coefficients being 1; specific values ​​can be calibrated on labeled samples through least squares fitting or grid search.

[0129] The intervals where the structural stability index meets the stability judgment condition are extracted as stable evolution intervals, and the intervals where the disturbance characteristic index is greater than the disturbance judgment threshold are extracted as mutation intervals, generating dynamic partitioning results.

[0130] Specifically, the dynamic partitioning results are generated as follows: traverse all relaxation time nodes along the relaxation time axis in ascending order of node index. For node segments where the structural stability indices of multiple consecutive nodes all meet the stability criteria, merge them into a continuous interval and denote it as a stable evolution interval. For node segments where the perturbation characteristic indices of multiple consecutive nodes all exceed the perturbation threshold, merge them into a continuous interval and denote it as a mutation interval. For cases where only a single node's perturbation characteristic index exceeds the perturbation threshold, define that single node itself as a mutation interval. Arrange all identified stable evolution intervals and mutation intervals in the order of their discrete index subscripts according to the start and end nodes to form a structured segmented description of the response ratio on the relaxation time axis, which is the dynamic partitioning result.

[0131] The stability determination condition can be set as a continuous interval formed by several nodes in which the structural stability index continuously exceeds the preset stability threshold. The preset stability threshold can be taken as the median or upper quartile of the distribution of structural stability index; the disturbance determination threshold can be taken as the upper quartile or 95th percentile of the distribution of disturbance characteristic index.

[0132] The dynamic partitioning results are compared with the positions of the overlapping peak intervals. The response ratios overlapping within the abrupt change intervals are numerically scaled to generate the reconstructed response ratios. The specific formula is as follows:

[0133] ,in, ;

[0134] in, For nodes Reconstruction response ratio at the location, Represents a node Response ratio at the location, This represents the adaptive numerical scaling factor determined by the perturbation characteristics. This represents the set of node indices that overlap within the mutation interval after comparison.

[0135] The value of the adaptive numerical scaling factor is negatively correlated with the corresponding node perturbation characteristic index: the larger the perturbation characteristic index, the smaller the numerical scaling factor, and the stronger the reduction in response ratio; the smaller the perturbation characteristic index, the closer the numerical scaling factor is to 1, and the smaller the change in response ratio. Specifically, it can be implemented using the inverse mapping of the perturbation characteristic index (e.g., using 1 plus the product of the adjustment sensitivity constant coefficient and the perturbation characteristic index as the denominator and 1 as the numerator) or the negative exponential mapping (e.g., using the negative of the product of the adjustment sensitivity constant coefficient and the perturbation characteristic index as the natural exponent). Its value range is constrained to the interval (0,1]. For nodes not in the abrupt change interval, the reconstructed response ratio remains unchanged.

[0136] The determination result of the deep coupling interval and the heterogeneous weight matrix are corrected by using the reconstruction response ratio. The fundamental frequency impedance vector is iteratively calculated based on the corrected heterogeneous weight matrix until the aliasing peak is separated into independent characteristic peaks.

[0137] Specifically, the determination result of the corrected deep coupling interval refers to: re-executing the deep coupling interval determination process in step S3 with the reconstructed response ratio instead of the original response ratio, including re-acquiring the fundamental frequency peak node and the ratio extreme value node, recalculating the curvature feature and Euclidean distance, and re-comparing with the preset curvature threshold and preset deviation threshold to obtain the corrected deep coupling interval. The correction of the heterogeneous weight matrix refers to: within the corrected deep coupling interval, performing the calculation of the statistical mean and variance and the construction of the penalty weight distribution curve in step S4 with the reconstructed response ratio, substituting the reconstructed response ratio of each node in the corrected deep coupling interval into the penalty weight distribution curve node by node to obtain the corresponding penalty weight, and arranging the standard penalty weight of the nodes outside the corrected deep coupling interval with it in node index order to form the corrected heterogeneous weight matrix. The iterative calculation of the fundamental frequency impedance vector based on the corrected heterogeneous weight matrix refers to: substituting the corrected heterogeneous weight matrix into the deconvolution objective function in step S2 to replace the standard penalty weight, and performing iterative solutions according to the iterative process in step S5 until the aliasing peak separation criterion or convergence condition is met;

[0138] In actual data processing, the response ratio itself may exhibit abnormal jumps at individual nodes due to numerical disturbances introduced during harmonic extraction. If such local disturbances are indiscriminately included in the construction of the heterogeneous weight matrix, they may affect the determination of the deep coupling interval. To identify the local disturbances in the response ratio itself and perform targeted numerical corrections, a structural stability index and a disturbance characteristic index based on multi-scale decomposition are introduced to perform structured partitioning and reconstruction of the response ratio.

[0139] With the above extended scheme, when the response ratio itself has local perturbations, the structured information obtained by multi-scale decomposition can be used to locate and numerically scale the perturbed nodes, so that the subsequent deep coupling interval determination and heterogeneous weight matrix constructed based on the response ratio are not affected by a small number of abnormal nodes. Alternatively, the numerical scaling of the response ratio in the abrupt interval can also be done by local weighted regression (LOWESS) smoothing, median filtering or spline interpolation.

[0140] Following the previous example, the obtained response ratio is decomposed into four levels using the Daubechies-4 wavelet basis. Three types of features are extracted from the decomposed detail coefficients and substituted into the weighted mapping calculation. The resulting structural stability index has a logarithmic relaxation time of approximately [missing information]. The values ​​in the vicinity are relatively high, corresponding to a stable evolution range; at the same scale, the perturbation characteristic index is approximately... A few nearby nodes exhibit spikes exceeding the judgment threshold; these nodes correspond to abrupt change intervals. Since these abrupt change intervals partially overlap with the aforementioned aliasing peak intervals, the response ratio for overlapping nodes is adjusted using an adaptive numerical scaling factor negatively correlated with the perturbation characteristic index (in this example, the scaling factor for the corresponding node is approximately...). The scaling process is performed to obtain the reconstructed response ratio. The deep coupling interval determination in S3 and the heterogeneous weight matrix construction in S4 are re-executed with the reconstructed response ratio. Then, the iterative operation in S5 is entered. The two independent characteristic peaks obtained are basically consistent with the results without perturbation correction in terms of position and amplitude. Moreover, the number of iterations required for iterative convergence is reduced compared with the result without perturbation correction.

[0141] Example 2:

[0142] In one embodiment of the present invention, which differs from the previous embodiment, the electronic device includes one or more processors and a memory.

[0143] A processor can be a central processing unit (CPU) or other form of processing unit with data processing and / or instruction execution capabilities, and can control other components in an electronic device to perform desired functions.

[0144] The memory may include one or more computer program products, which may include various forms of computer-readable storage media, such as volatile memory and / or non-volatile memory. Volatile memory may include, for example, random access memory (RAM) and / or cache memory. Non-volatile memory may include, for example, read-only memory (ROM), hard disk, flash memory, etc.

[0145] In one example, the electronic device may also include input devices and output devices, which are interconnected via a bus system and / or other forms of connection mechanisms (not shown). In addition, depending on the specific application, the electronic device may include any other suitable components.

[0146] Example 3:

[0147] Embodiments of this application may also be computer-readable storage media storing computer program instructions thereon, which, when executed by a processor, cause the processor to perform the steps described in the "Embodiment 1" section of this specification according to the various embodiments of this application.

[0148] Computer-readable storage media may take the form of any combination of one or more readable media. A readable medium may be a readable signal medium or a readable storage medium. A readable storage medium may, for example, include, but is not limited to, electrical, magnetic, optical, electromagnetic, infrared, or semiconductor systems, apparatuses, or devices, or any combination thereof. More specific examples of readable storage media (a non-exhaustive list) include: electrical connections having one or more wires, portable disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fibers, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof.

[0149] The basic principles of this application have been described above with reference to specific embodiments. However, it should be noted that the advantages, benefits, and effects mentioned in this application are merely examples and not limitations, and should not be considered as essential features of each embodiment of this application. Furthermore, the specific details disclosed above are for illustrative and facilitative purposes only, and are not limitations. These details do not restrict the application from being implemented using the specific details described above.

[0150] The block diagrams of devices, apparatuses, devices, and systems involved in this application are merely illustrative examples and are not intended to require or imply that they must be connected, arranged, or configured in the manner shown in the block diagrams. As those skilled in the art will recognize, these devices, apparatuses, devices, and systems can be connected, arranged, and configured in any manner. Words such as “comprising,” “including,” “having,” etc., are open-ended terms meaning “including but not limited to,” and are used interchangeably with them. The terms “or” and “and” as used herein refer to the terms “and / or,” and are used interchangeably with them unless the context clearly indicates otherwise. The term “such as” as used herein refers to the phrase “such as but not limited to,” and is used interchangeably with it.

[0151] It should also be noted that in the apparatus, equipment, and methods of this application, the components or steps can be disassembled and / or recombined. These disassemblies and / or recombinations should be considered as equivalent solutions of this application.

[0152] The above description of the disclosed aspects is provided to enable any person skilled in the art to make or use this application. Various modifications to these aspects will be readily apparent to those skilled in the art, and the general principles defined herein can be applied to other aspects without departing from the scope of this application. Therefore, this application is not intended to be limited to the aspects shown herein, but rather to be accorded the widest scope consistent with the principles and novel features disclosed herein.

[0153] The above description has been given for purposes of illustration and description. Furthermore, this description is not intended to limit the embodiments of this application to the forms disclosed herein. Although numerous exemplary aspects and embodiments have been discussed above, those skilled in the art will recognize certain variations, modifications, alterations, additions, and sub-combinations thereof.

Claims

1. A method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy, characterized in that, include: An AC signal with progressively increasing amplitude is applied to the interface under test, and the fundamental frequency impedance vector and harmonic impedance vector are extracted. The fundamental frequency impedance vector and the harmonic impedance vector are deconvolved using preset standard penalty weights to obtain the fundamental frequency relaxation vector and the harmonic relaxation vector. The amplitudes of the two are then divided by the nodes to obtain the response ratio. Identify aliasing peak intervals in the fundamental frequency relaxation vector where the information entropy is greater than an entropy threshold, wherein the aliasing peak interval contains aliasing peaks. The aliasing peak interval is confirmed as a deeply coupled interval only when the peak position of the aliasing peak overlaps with the extreme point of the response ratio, including: The zero points of the first derivative of the fundamental frequency relaxation vector and the zero points of the first derivative of the response ratio within the aliasing peak interval are obtained respectively to obtain the fundamental frequency peak node and the ratio extreme value node. Extract the curvature features of the response ratio in the neighborhood of the fundamental frequency peak node, and calculate the Euclidean distance between the fundamental frequency peak node and the ratio extreme value node; The aliasing peak interval is confirmed as a deep coupling interval only when the curvature feature is greater than a preset curvature threshold and the Euclidean distance is less than a preset deviation threshold. A heterogeneous weight matrix is ​​generated based on the response ratio within the deep coupling interval: a penalty weight less than the standard penalty weight is assigned to nodes where the response ratio is greater than a preset ratio judgment value, and a penalty weight greater than or equal to the standard penalty weight is assigned to nodes that approach zero. The heterogeneous weight matrix is ​​used to replace the standard penalty weight. Based on the replaced penalty weight, the fundamental frequency impedance vector is iteratively calculated to generate an updated fundamental frequency relaxation vector until the aliasing peak is separated into independent characteristic peaks, and the decoupled charge transfer resistance and double-layer capacitance are output.

2. The method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy according to claim 1, characterized in that, After obtaining the fundamental frequency relaxation vector, the following steps are also included: Acquire the instantaneous phase characteristics, amplitude drift characteristics, and frequency dependence characteristics of the interface under test under temperature change conditions or electrochemical excitation conditions; The instantaneous phase feature, the amplitude drift feature, and the frequency dependence feature are weighted and summed to generate a modulation vector; The modulation vector and the fundamental frequency relaxation vector are fused node by node. High coupling intervals in the fused data with gradient change rate greater than gradient determination threshold are identified. The signal features in the high coupling intervals are numerically amplified to generate enhanced relaxation vectors. The response ratio and the heterogeneous weight matrix are updated based on the enhanced relaxation vector.

3. The method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy according to claim 2, characterized in that, The generation of the enhanced relaxation vector includes: Extract the instantaneous phase features corresponding to each relaxation time node within the high coupling interval; Calculate the absolute phase deviation between the instantaneous phase characteristic and the ideal capacitor phase reference value; Construct an asymmetric amplification factor that is positively correlated with the absolute phase deviation; The magnitude of the fundamental frequency relaxation vector within the high coupling interval is multiplied node by node by the asymmetric amplification coefficient to generate the enhanced relaxation vector.

4. The method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy according to claim 1, characterized in that, After obtaining the response ratio, the following is also included: The response ratio is decomposed into multiple scales to extract local fluctuation features, phase coupling features and energy distribution features at different scales; The local wave characteristics, the phase coupling characteristics, and the energy distribution characteristics are weighted and mapped to construct structural stability index and disturbance characteristic index of response ratio, respectively. The intervals where the structural stability index meets the stability judgment condition are extracted as stable evolution intervals, and the intervals where the disturbance feature index is greater than the disturbance judgment threshold are extracted as mutation intervals, generating dynamic partitioning results. The dynamic partitioning results are compared with the positions of the aliasing peak intervals, and the response ratios overlapping the abrupt change intervals are numerically scaled to generate the reconstructed response ratios. The determination result of the deep coupling interval and the heterogeneous weight matrix are corrected using the reconstruction response ratio. The fundamental frequency impedance vector is iteratively calculated based on the corrected heterogeneous weight matrix until the aliasing peak is separated into independent characteristic peaks.

5. The method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy according to claim 1, characterized in that, The extraction of the curvature features includes: Extract the adjacent local minimum nodes of the response ratio on both sides of the fundamental frequency peak node, and define the continuous interval between the two local minimum nodes as the neighborhood; Calculate the second-order difference value of the response ratio at the fundamental frequency peak node, and extract the absolute value of the second-order difference value as the absolute sharpness index. Numerical integration is performed on the response ratio within the neighborhood to generate a local evolution energy index. Calculate the ratio of the absolute sharpness index to the local evolution energy index, and use the ratio as the curvature feature.

6. The method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy according to claim 1, characterized in that, At nodes where the response ratio is greater than a preset ratio determination value, a penalty weight less than the standard penalty weight is assigned; at nodes approaching zero, a penalty weight greater than or equal to the standard penalty weight is assigned. Specifically, this includes: Obtain the statistical mean and statistical variance of the response ratio within the deep coupling interval; The statistical mean is used as the numerical switching center of the penalty weight, and the positively correlated statistical variance is used as the switching steepness factor. A penalty weight distribution curve is constructed based on the numerical switching center and the switching steepness factor. Using the penalty weight distribution curve, the response ratio within the deep coupling interval is converted into a corresponding penalty weight node by node.

7. The method for analyzing interfacial reaction kinetic parameters based on electrochemical impedance spectroscopy according to claim 6, characterized in that, The acquisition of the statistical mean and statistical variance includes: Perform node-by-node difference operations on the response ratio within the deep coupling interval to extract the local gradient feature values ​​of each node; Construct a continuity confidence coefficient that is negatively correlated with the absolute value of the local gradient feature value; Using the continuity confidence coefficient as the weight, a weighted average is calculated on the response ratio within the deep coupling interval to obtain the weighted center value, and the weighted center value is determined as the statistical mean. Calculate the squared deviation between the response ratio of each node and the statistical mean, use the continuity confidence coefficient to perform a weighted average of the squared deviations to obtain the weighted dispersion, and determine the weighted dispersion as the statistical variance.

8. An electronic device comprising a memory and a processor, characterized in that: The memory is used to store computer-executable instructions, and the processor is used to execute the computer-executable instructions, which, when executed by the processor, implement the steps of the method as described in any one of claims 1 to 7.

9. A computer storage medium storing computer-executable instructions thereon, characterized in that: When the computer-executable instructions are executed by a processor, they implement the steps of the method as described in any one of claims 1 to 7.

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