Research on Scale Factor Synthesis Modeling Methods for 8/20 μs Impulse Waveforms

Abstract

The synthetic model of the impulse scale factor is the foundation for tracing impulse waveform peaks to AC measurement standards. However, existing research has primarily focused on double-exponential 1.2/50 μs impulse waveforms, while for 8/20 μs impulse current waveforms—characterized as exponentially damped sinusoidal oscillations with pronounced spectral broadening and oscillatory features—conventional models fail to provide accurate characterization, leading to significantly increased synthetic errors. Based on Fourier transform theory, this study derives a multi-parameter mapping relationship between the impulse scale factor and AC scale factors. A fractional-order error suppression method and frequency band optimization strategy are proposed, establishing a synthetic impulse scale factor model suitable for 8/20 μs waveforms that significantly extends the applicability of existing models. Simulation results demonstrate that with the fractional-order parameter z = 0.1, the relative error of μ decreases by a factor of 12 times compared to existing models; meanwhile, the optimized frequency band division reduces data acquisition workload by 39% while maintaining measurement accuracy. This research defines the applicability boundaries of scale factor synthetic models under 8/20 μs impulse waveforms, providing theoretical support and practically efficient modeling methods for the traceability of impulse peak values.

1. Introduction

Impulse current waveforms are widely used in military, industrial, and energy sectors. In military applications, pulsed power technology serves as the core driving basis for systems such as electromagnetic launch and high-power microwaves [1,2]. In the industrial field, high-voltage impulses are employed in processes like mineral separation and rock fragmentation, demonstrating high-efficiency energy conversion characteristics [3,4,5,6]. Within power systems, impulse waveforms are extensively applied in critical areas such as equipment testing, condition monitoring, and lightning protection. According to relevant IEC and IEEE standards, typical impulse current waveforms primarily include the 1/20 μs steep wave, 8/20 μs lightning current, 30/80 μs switching surge, and 4/10 μs high impulse [7]. Among these, the 8/20 μs impulse current, serving as a standard waveform for simulating natural lightning strikes and testing the residual voltage performance of surge protective devices, holds significant importance in power system insulation coordination and lightning protection design [8,9,10,11].

Accurate measurement of impulse currents is a prerequisite for evaluating equipment performance and ensuring system safety. Consequently, establishing a reliable traceability system—linking impulse measurement results to national or international standards through an unbroken calibration chain—is essential [12]. Current traceability methods for the impulse scale factor can be primarily categorized into three classes: the standard source/component-based method, the single-frequency method, and the multi-frequency synthesis method. These methods exhibit distinct characteristics in terms of principle, accuracy, and waveform adaptability, as summarized in

Table 1. Comparison of traceability methods for impulse current measuring systems.

Category	Representative Works	Core Methodology	Advantages	Limitations
Standard Source/Component-Based	[13,14,15,16]	Constructs a measurement system using a standard impulse current source or independently calibrated components (e.g., shunts, digitizers), achieving traceability via direct comparison or the product of component scale factors.	Intuitive concept and relatively straightforward implementation. Capable of providing standard waveforms with high repeatability [13].	Developing high-amplitude standard sources is costly and results in complex systems [14]. Often assumes the impulse scale factor equals the DC/low-frequency scale factor, failing to fully consider the spectral characteristics of the waveform [15]. Traceability accuracy is limited by the inherent accuracy of the standard source/components, making further improvement difficult [16].
Single-Frequency Method	[17]	Assumes the measuring system has an ideally flat frequency response, equating the impulse scale factor to the AC scale factor at a single frequency (e.g., power frequency).	Simple to implement.	Neglects the system’s non-ideal frequency response and the broad spectrum of the impulse waveform, potentially leading to significant deviations [17].
Multi-Frequency Synthesis Method	[18,19]	Models the impulse scale factor as a weighted synthesis of AC scale factors at multiple frequencies based on Parseval’s theorem and energy spectral density.	Theoretically more rigorous, significantly improving traceability accuracy [18,19]. Establishes a direct link between impulse scale factor and AC standards through frequency domain analysis.	Existing research primarily focuses on standard lightning impulse voltages (e.g., 1.2/50 μs), with insufficient systematic study and model optimization tailored for the parameter sensitivity of the 8/20 μs impulse current waveform [18,19].

below.

A systematic review of the existing literature reveals that the standard source/component-based and single-frequency methods face inherent bottlenecks when dealing with complex waveforms and high-accuracy requirements due to their fundamental assumptions. The multi-frequency synthesis method represents the state-of-the-art traceability concept, offering the highest potential accuracy and theoretical universality. However, as evidenced by the works of [18,19], despite its successful application to standard lightning impulse voltages, systematic research and model optimization specifically for the 8/20 μs impulse current waveform remain relatively scarce. Since the parameters of the mathematical model for the 8/20 μs waveform are highly susceptible to disturbances, directly applying synthesis models established for other waveforms (e.g., 1.2/50 μs) introduces considerable errors, highlighting the limitations in the waveform adaptability of current models.

To this end, this paper presents a systematic construction of a synthetic impulse scale factor model tailored for the 8/20 μs impulse current waveform, along with a clear delineation of its applicable scope. The paper is structured as follows: Section 2 elaborates on the synthesis workflow of the model and establishes a multi-parameter analytical expression through theoretical derivation. In Section 3, the key parameter μ in the expression is optimized based on fractional-order theory to reduce model application error. Section 4 illustrates the model’s application via a specific case study and provides a comprehensive analysis comparing the synthesis accuracy before and after model optimization, as well as under different frequency band division strategies. Finally, Section 5 concludes the paper and outlines potential directions for future research.

The main contributions of this work are summarized as follows:

• A tailored synthetic model for 8/20 μs waveforms: Systematic derivation and establishment of a synthetic impulse scale factor model specifically designed for the exponentially damped sinusoidal 8/20 μs impulse current waveform, addressing a critical gap in the traceability of non-double-exponential impulses.
• Fractional-order optimization for error suppression: Introduction of a novel bilevel optimization framework to determine the optimal fractional-order parameter z, effectively suppressing the model’s inherent nonlinear error by an order of magnitude compared to conventional approaches.
• An efficient frequency band division strategy: Development of an optimized frequency band division strategy that reduces the required calibration frequency points by approximately 39% while maintaining synthesis accuracy, significantly enhancing practical measurement efficiency.

2. Synthetic Model of Impulse Scale Factor

To achieve high-precision metrology for 8/20 μs impulse current waveforms, establishing a mapping relationship between the impulse scale factor and standard AC scale factors is essential, where constructing a synthetic impulse scale factor model constitutes the pivotal procedure. This section focuses on synthetic modeling, proposing a systematic method that initiates from raw signals and derives the model through frequency-energy domain fusion, along with optimization analysis of key parameters.

As shown in Figure 1, the overall impulse scale factor synthesis process includes: (1) Applying Fourier transforms to input and output impulse waveforms to obtain their spectra, computing the ratio of their spectral amplitudes; (2) Leveraging Parseval’s theorem to correlate time-domain signal energy with frequency-domain energy spectra, introducing energy proportion as a weighting factor to build a mapping model between the impulse scale factor and various AC scale factors; (3) Eliminating irrelevant terms introduced by signal amplitudes (A/A′) through a specific fractional-order structure, allowing the model to exclusively reflect intrinsic system properties; (4) Conducting parameter space exploration and optimization of the key parameter μ to analyze its impact on scale factor accuracy and system adaptability, obtaining the optimal model form. Through this optimization process, optimal fractional-order parameters are determined to establish the theoretical basis for tracing impulse scale factors to AC standard quantities.

The construction of the synthetic impulse scale factor model depends not only on the amplitude–frequency characteristics of the measuring system but is also significantly influenced by the time-domain structure and spectral distribution of the impulse waveform itself. Compared to the internationally standardized 1.2/50 μs double-exponential voltage waveform, the more prevalent 8/20 μs impulse current waveform in power systems exhibits distinct functional forms, energy concentration, and frequency composition, necessitating a dedicated mathematical model for this waveform is imperative.

The 8/20 μs impulse current waveform is generated by an RLC circuit operating in an underdamped state [20,21], and its analytical expression is as follows:

$$x\left(t\right)=A\cdot {\mathrm{e}}^{\alpha t}sin\left(\beta t\right) \left(\alpha <0,\beta >0\right)$$

(1)

$$\alpha =-{\displaystyle \frac{R}{2L}}$$

(2)

$$x\left(t\right)=A\cdot {\mathrm{e}}^{\alpha t}sin\left(\beta t\right) \left(\alpha <0,\beta >0\right)$$

(3)

where A is a scaling factor for the peak adjustment factor of the input waveform, with its unit in amperes (A); R, L, and C represent the resistance, inductance, and capacitance in the equivalent circuit of the generator, respectively. As indicated by their expressions, both α and β are reciprocals of time constants, thus exhibiting the dimension of angular frequency. For 8/20 μs characteristics, α = 41,628.474 and β = 120,023.328. The response of measuring system is simplified as:

$$y\left(t\right)=A^{\prime }\cdot {\mathrm{e}}^{{\alpha }^{\prime }t}sin\left({\beta }^{\prime }t\right) \left({\alpha }^{\prime }<0,{\beta }^{\prime }>0\right)$$

(4)

where A′ is the peak adjustment factor of output waveforms, with its unit in volts (V), α′ and β′ are model parameters typically obtained by fitting measured data. Differentiating (1) and (2) and setting derivatives to zero yields the peak times t_p and t_p′ for input and output waveforms, respectively. Substituting these into the original equations gives the peaks [18]:

$$x_{\mathrm{p}}={\displaystyle \frac{A\beta {\mathrm{e}}^{\frac{\alpha }{\beta }\cdot arctan\left(\frac{-\beta }{\alpha }\right)}}{\sqrt{{\alpha }^2+{\beta }^2}}}$$

(5)

$$y_{\mathrm{p}}={\displaystyle \frac{A^{\prime }{\beta }^{\prime }{\mathrm{e}}^{\frac{{\alpha }^{\prime }}{{\beta }^{\prime }}\cdot arctan\left(\frac{-{\beta }^{\prime }}{{\alpha }^{\prime }}\right)}}{\sqrt{{{\alpha }^{\prime }}^2+{{\beta }^{\prime }}^2}}}$$

(6)

The impulse scale factor is defined as the ratio of output to input peak values:

$$k={\displaystyle \frac{A}{A^{\prime }}}\cdot {\displaystyle \frac{\beta }{{\beta }^{\prime }}}\cdot {\displaystyle \frac{{\mathrm{e}}^{\frac{\alpha }{\beta }\cdot arctan\left(\frac{-\beta }{\alpha }\right)}}{{\mathrm{e}}^{\frac{{\alpha }^{\prime }}{{\beta }^{\prime }}\cdot arctan\left(\frac{-{\beta }^{\prime }}{{\alpha }^{\prime }}\right)}}}\cdot {\displaystyle \frac{\sqrt{{{\alpha }^{\prime }}^2+{{\beta }^{\prime }}^2}}{\sqrt{{\alpha }^2+{\beta }^2}}}$$

(7)

To establish a theoretical mapping from the impulse scale factor to AC scale factors, Fourier transforms are introduced to convert time-domain waveforms to frequency-domain. The input and output waveform spectra are respectively

$$X(j\omega )={\displaystyle {\int }_0^{\infty }x\left(t\right)\cdot {\mathrm{e}}^{jwt}}dt={\displaystyle \frac{A\beta }{{\beta }^2+{\left(\alpha -jw\right)}^2}}$$

(8)

$$Y(j\omega )={\displaystyle {\int }_0^{\infty }y\left(t\right)\cdot {\mathrm{e}}^{jwt}}dt={\displaystyle \frac{A^{\prime }{\beta }^{\prime }}{{{\beta }^{\prime }}^2+{\left({\alpha }^{\prime }-jw\right)}^2}}$$

(9)

According to (8) and (9), the AC scale factor can be expressed as:

$$\begin{array}{cc}\hfill k_{AC}(w)& =\left|{\displaystyle \frac{\frac{A\beta }{{\beta }^2+{\left(\alpha -jw\right)}^2}}{\frac{A^{\prime }{\beta }^{\prime }}{{{\beta }^{\prime }}^2+{\left({\alpha }^{\prime }-jw\right)}^2}}}\right|={\displaystyle \frac{\left|A\right|\cdot \beta \cdot \sqrt{{\left({{\alpha }^{\prime }}^2+{{\beta }^{\prime }}^2-w^2\right)}^2+4{{\alpha }^{\prime }}^2{w}^2}}{\left|A^{\prime }\right|\cdot {\beta }^{\prime }\cdot \sqrt{{\left({\alpha }^2+{\beta }^2-w^2\right)}^2+4{\alpha }^2{w}^2}}}\hfill \\ & ={\displaystyle \frac{\left|A\right|}{\left|A^{\prime }\right|}}\cdot p\left(\alpha ,\beta ,{\alpha }^{\prime },{\beta }^{\prime },w\right)\hfill \end{array}$$

(10)

At the low frequency w_low (which can be taken as the power frequency), k_AC(w) is essentially constant, and the dependence on w can be neglected. This yields:

$$\begin{array}{cc}\hfill k_{AC}(w_{low})& ={\displaystyle \frac{\left|A\right|\cdot \beta \cdot \sqrt{{\left({{\alpha }^{\prime }}^2+{{\beta }^{\prime }}^2-{w_{low}}^2\right)}^2+4{{\alpha }^{\prime }}^2{w_{low}}^2}}{\left|A^{\prime }\right|\cdot {\beta }^{\prime }\cdot \sqrt{{\left({\alpha }^2+{\beta }^2-{w_{low}}^2\right)}^2+4{\alpha }^2{w_{low}}^2}}}\hfill \\ & ={\displaystyle \frac{\left|A\right|}{\left|A^{\prime }\right|}}\cdot p\left(\alpha ,\beta ,{\alpha }^{\prime },{\beta }^{\prime }\right)\hfill \end{array}$$

(11)

It should be noted that k depends solely on system characteristics and is independent of input amplitude A. Meanwhile, output A′ varies linearly with input A, meaning the ratio A′/A only reflects amplitude scaling, failing to reveal frequency-dependent effects on the scale factor, and thus should be eliminated. Although (7) and (11) partially constrain A′/A and could be used to relate impulse and low-frequency AC scale factors, they inadequately capture frequency-dependent system responses. Additional frequency-domain constraints are necessary to comprehensively model spectral influences on the impulse scale factor.

Notably, impulse peak values (A, A′) quadratically increase total signal energy. To characterize the impulse waveform in the frequency dimension, energy is adopted as a unified physical quantity. The energy of the input waveform and that of the output waveform are given by:

$$W_x={\displaystyle \frac{1}{2\mathsf{\pi }}}{\displaystyle {\int }_{-\infty }^{\infty }{\left|X\left(jw\right)\right|}^2}dw=-{\displaystyle \frac{A^2{\beta }^2}{4{\alpha }^3+4\alpha {\beta }^2}}$$

(12)

$$W_y={\displaystyle \frac{1}{2\mathsf{\pi }}}{\displaystyle {\int }_{-\infty }^{\infty }{\left|Y\left(jw\right)\right|}^2}dw=-{\displaystyle \frac{{A^{\prime }}^2{{\beta }^{\prime }}^2}{4{{\alpha }^{\prime }}^3+4{\alpha }^{\prime }{{\beta }^{\prime }}^2}}$$

(13)

In a similar manner, A/A′ is derived by computing the energy ratio:

$${\displaystyle \frac{W_x}{W_y}}={\displaystyle \frac{A^2}{{A^{\prime }}^2}}\cdot {\displaystyle \frac{{\beta }^2}{{{\beta }^{\prime }}^2}}\cdot {\displaystyle \frac{{{\alpha }^{\prime }}^3+{\alpha }^{\prime }{{\beta }^{\prime }}^2}{{\alpha }^3+\alpha {\beta }^2}}={\displaystyle \frac{A^2}{{A^{\prime }}^2}}\cdot q\left(\alpha ,\beta ,{\alpha }^{\prime },{\beta }^{\prime }\right)$$

(14)

Assuming A, A′ > 0, substituting (11) and (14) into (7) eliminates the ratio A′/A, which yields a simplified expression for k as given in (15). The complete derivation is provided in (A1) of the Appendix A.

$$k={\left[k_{AC}(w_{low})\right]}^z\cdot {\left[{\displaystyle \frac{W_x}{W_y}}\right]}^{{\displaystyle \frac{1-z}{2}}}\cdot \mu \left(\alpha ,\beta ,{\alpha }^{\prime },{\beta }^{\prime },z\right)$$

(15)

where z is a fractional-order exponent to be optimized, reflecting the nonlinear influence of system energy transfer efficiency on μ. Its value will be determined via simulation optimization. However, a remaining issue is that the energy of the input impulse is unknown, so the ratio W_x/W_y in (15) needs to be further determined. According to the concept of infinitesimal elements, W_x and W_y can be expressed as (16) and (17), respectively. This allows the calculation of W_x/W_y

$$\begin{array}{c}{W}_x={\displaystyle \frac{1}{2\mathsf{\pi }}}{\displaystyle {\int }_{-\infty }^{\infty }|X(jw){|}^2}dw={\displaystyle \frac{1}{\mathsf{\pi }}}{\displaystyle {\int }_0^{\infty }|X(jw){|}^2}dw\hfill \\ \hspace{1em} ={\displaystyle \sum _i^n\frac{{\left|X(j{w}_{i-1})\right|}^2+{\left|X(j{w}_i)\right|}^2}{2}\times \frac{2\mathsf{\pi }{f}_i-2\mathsf{\pi }{f}_{i-1}}{\mathsf{\pi }}}={\displaystyle \sum _i^n\left[{\left|X(j{w}_{i-1})\right|}^2+{\left|X(j{w}_i)\right|}^2\right]\times \left(f_i-f_{i-1}\right)}\hfill \end{array}$$

(16)

$$\begin{array}{c}{W}_y={\displaystyle \frac{1}{2\mathsf{\pi }}}{\displaystyle {\int }_{-\infty }^{\infty }|Y(jw){|}^2}dw={\displaystyle \frac{1}{\mathsf{\pi }}}{\displaystyle {\int }_0^{\infty }|Y(jw){|}^2}dw\hfill \\ \hspace{1em} ={\displaystyle \sum _i^n\frac{{\left|Y(j{w}_{i-1})\right|}^2+{\left|Y(j{w}_i)\right|}^2}{2}\times \frac{2\mathsf{\pi }{f}_i-2\mathsf{\pi }{f}_{i-1}}{\mathsf{\pi }}}={\displaystyle \sum _i^n\left[{\left|Y(j{w}_{i-1})\right|}^2+{\left|Y(j{w}_i)\right|}^2\right]\times \left(f_i-f_{i-1}\right)}\hfill \end{array}$$

(17)

$$\begin{array}{c}{\displaystyle \frac{W_y}{W_x}}={\displaystyle \frac{{\displaystyle \sum _i^n\left[{\left|Y(j{w}_{i-1})\right|}^2+{\left|Y(j{w}_i)\right|}^2\right]\times \left(f_i-f_{i-1}\right)}}{{\displaystyle \sum _i^n\left[{\left|X(j{w}_{i-1})\right|}^2+{\left|X(j{w}_i)\right|}^2\right]\times \left(f_i-f_{i-1}\right)}}}\hfill \\ \hspace{1em} ={\displaystyle \sum _{i=1}^n\frac{{\left|X(j{w}_i)\right|}^2\times \left(f_i-f_{i-1}\right)}{{\left|X(j{w}_1)\right|}^2\times \left(f_1-f_0\right)+{\left|X(j{w}_2)\right|}^2\times \left(f_2-f_1\right)+\cdots +{\left|X(j{w}_n)\right|}^2\times \left(f_n-f_{n-1}\right)\cdots }}\cdot {\displaystyle \frac{{\left|Y(j{w}_i)\right|}^2}{{\left|X(j{w}_i)\right|}^2}}\hfill \\ \hspace{1em} ={\displaystyle \sum _{i=1}^n{S}_{X(i)}\cdot {\left[\frac{Y(j{w}_i)}{X(j{w}_i)}\right]}^2}={\displaystyle \sum _i^n\frac{S_{X(i)}}{{k_i}^2}}\hfill \end{array}$$

(18)

where S_X(i) is the energy proportion within different bands, and k_i is the AC scale factor at different frequencies. Equation (18) integrates system frequency response with signal spectral distribution, providing frequency-domain support for the synthetic impulse scale factor model.

In summary, Equations (15)–(18) form the core theoretical framework of the synthetic impulse scale factor model based on frequency-domain energy distribution, laying the foundation for subsequent parameter optimization and frequency band division.

3. Determination of z

According to (15), when the time constants of the output waveform equal those of the input (α′ = α, β′ = β), μ = 1, and the model introduces no additional error. Deviation of α′ and β′ from α and β causes nonlinear scaling of μ with z, affecting the accuracy of the calculated impulse scale factor k. To suppress model uncertainty, systematic analysis and optimization of z are essential.

Let:

$$\phi \left[k_{AC}(w),z\right]={\left[{\displaystyle \frac{k_{AC}(w_{low})}{\sqrt{W_x/W_y}}}\right]}^z,k_{AC}(w_{low})\in k_{AC}(w)$$

(19)

Equation (15) can be rewritten as:

$$k={\left[{\displaystyle \frac{W_x}{W_y}}\right]}^{{\displaystyle \frac{1}{2}}}\cdot \phi \left[k_{AC}(w),z\right]\cdot \mu \left(\alpha ,\beta ,{\alpha }^{\prime },{\beta }^{\prime },z\right)$$

(20)

To ensure the accuracy of k, its uncertainty should be as small as possible. The uncertainty is expressed as follows:

$$u\left(k\right)={\left[{\displaystyle \frac{W_x}{W_y}}\right]}^{{\displaystyle \frac{1}{2}}}\cdot \sqrt{{\displaystyle \sum _w{\left[\frac{\partial \phi }{\partial k_{AC}}\cdot u\left(k_{AC}\right)\cdot \mu \right]}^2}+{\left[{\displaystyle \frac{\partial \phi }{\partial z}}\cdot u\left(z\right)\cdot \mu \right]}^2+{\left[\phi \cdot {\displaystyle \frac{\partial \mu }{\partial z}}\cdot u\left(z\right)\right]}^2}$$

(21)

where ${\displaystyle \frac{\partial \phi }{\partial z}}={\left[{\displaystyle \frac{k_{AC}(w_{low})}{\sqrt{W_x/W_y}}}\right]}^z\cdot \ln \left[{\displaystyle \frac{k_{AC}(w_{low})}{\sqrt{W_x/W_y}}}\right]$. Since $k_{AC}(w_{low})$ and $\sqrt{W_x/W_y}$ are of the same order of magnitude and very close to each other, ${\displaystyle \frac{\partial \phi }{\partial z}}$ approximates zero.

Meanwhile, given the high accuracy of the calibration setup, $u\left(k_{AC}\right)$ is negligible. Therefore:

$$u\left(k\right)\approx {\left[{\displaystyle \frac{W_x}{W_y}}\right]}^{{\displaystyle \frac{1}{2}}}\cdot \phi \cdot {\displaystyle \frac{\partial \mu }{\partial z}}\cdot u\left(z\right)$$

(22)

The direct computation of ${\displaystyle \frac{\partial \mu }{\partial z}}\cdot u\left(z\right)$ presents difficulties; however, the problem can be transformed into an investigation of the maximum error. The objective is to minimize the maximum deviation of μ from unity. Regarding the value of z, it can be seen that when the measuring system has infinite bandwidth, ${\displaystyle \frac{\beta \cdot \sqrt{{\left({{\alpha }^{\prime }}^2+{{\beta }^{\prime }}^2-{w_{low}}^2\right)}^2+4{{\alpha }^{\prime }}^2{w_{low}}^2}}{{\beta }^{\prime }\cdot \sqrt{{\left({\alpha }^2+{\beta }^2-{w_{low}}^2\right)}^2+4{\alpha }^2{w_{low}}^2}}}$, ${\displaystyle \frac{{\beta }^2}{{{\beta }^{\prime }}^2}}\cdot {\displaystyle \frac{{{\alpha }^{\prime }}^3+{\alpha }^{\prime }{{\beta }^{\prime }}^2}{{\alpha }^3+\alpha {\beta }^2}}$, and ${\displaystyle \frac{\beta }{{\beta }^{\prime }}}\cdot {\displaystyle \frac{{\mathrm{e}}^{\frac{\alpha }{\beta }\cdot arctan\left(\frac{-\beta }{\alpha }\right)}}{{\mathrm{e}}^{\frac{{\alpha }^{\prime }}{{\beta }^{\prime }}\cdot arctan\left(\frac{-{\beta }^{\prime }}{{\alpha }^{\prime }}\right)}}}\cdot {\displaystyle \frac{\sqrt{{{\alpha }^{\prime }}^2+{{\beta }^{\prime }}^2}}{\sqrt{{\alpha }^2+{\beta }^2}}}$ in μ are all equal to 1. In practice, when the relative errors of the output waveform’s time parameters compared to the input waveform are small, these three terms also approach 1. The role of z is to constrain the value of μ within a small range through appropriate mapping. Therefore, in the subsequent optimization, to avoid inappropriate scaling, z can be constrained within the range of ±1. Therefore, the optimization model can be formulated as follows

$$\begin{array}{l}\underset{z}{\min } \underset{{\alpha }^{\prime },{\beta }^{\prime }}{\max } \left|\mu \left(\alpha ,\beta ,{\alpha }^{\prime },{\beta }^{\prime },z\right)-1\right|\\ s.t. \left\{\begin{array}{c}z \in [-1,1]\\ ({\alpha }^{\prime },{\beta }^{\prime })\in \Omega \end{array}\right.\end{array}$$

(23)

This is inherently a bilevel optimization problem: the inner level finds α’ and β’ combinations maximizing μ error for a given z, while the outer level traverses z to minimize this maximum error. To simplify, a combined strategy of outer traversal and inner exhaustive search is adopted. The specific optimization procedure is illustrated in Figure 2, which consists of the following key steps: First, the inner function is defined as the objective function. Next, the feasible regions of parameters α′ and β′ are converted into linear constraints. The fractional order z is then assigned a defined range, and the maximum values of the objective function under different z values are obtained through traversal. Finally, the z value that minimizes the maximum objective function value is identified as the optimal fractional order. To improve readability, the pseudocode is additionally included in Appendix A.2.

Referring to [19], errors in output waveform T₁ and T₂ relative to input are assumed not exceeding 1%. Figure 3a,b illustrates the corresponding increase in the time parameters T₁ and T₂ as α′ increases or β′ decreases. The color bars on the right indicate that when T₁ and T₂ fall within the green region, their relative errors do not exceed 0.25%. As the color gradient intensifies, it reflects the growing influence of α′ and β′ variations on the stability of T₁ and T₂. Moreover, within the 1% error margin for T₁ and T₂, the feasible region Ω formed by α′ and β′ approximates a parallelogram, providing a clear parameter range for subsequent optimization.

Traversal optimization using MATLAB yields maximum relative errors for different z. As shown in Figure 4, at z = –1, the derived model aligns with the double-exponential impulse scale factor model. For 1.2/50 μs impulses, μ-induced relative error is below 0.05%, negligible for k. For 8/20 μs impulses, however, this error rises to 6%. Analysis demonstrates that confining z to the interval [–0.04, 0.16] reduces the relative error to 0.6%, confirming the necessity of optimization. The optimal value z = 0.1 achieves a relative error no greater than 0.5%. To further suppress errors below 0.1%, deviations in output waveform parameters (T₁, T₂) relative to the input must be controlled within 0.2%.

Measurements using a self-integrating Rogowski coil (modeled as a bandpass filter) are performed:

$$G\left(s\right)={\displaystyle \frac{\lambda s}{\left(s+w_l\right)\left(s+w_u\right)}} \left(0<w_l<w_u\right)$$

(24)

where w_l and w_u represent the two zeros of the band-pass filter, rather than the −3 dB cutoff frequencies, and λ is an undetermined parameter. With w_l set to 10 × 2π rad/s and w_u at 10⁶ × 2π rad/s, the resulting output waveform is depicted in Figure 5. The relative errors in the output time parameters T₁ and T₂ are 1.60 × 10⁻³ and 6.25 × 10⁻⁵, respectively. Notably, the relative error of T₁ approaches 2 × 10⁻³, indicating that in practical applications, to preserve model accuracy, the lower and upper cutoff frequencies of the measurement device should not fall below 10 Hz and 1 MHz, respectively. When w_l = 0, the band-pass filter effectively becomes a low-pass filter.

Notably, reducing w_l or increasing w_u further decreases errors in T₁ and T₂, thereby improving synthetic impulse scale factor accuracy.

4. Frequency Band Division Strategy

According to (15) and (18), synthetic accuracy depends not only on amplitude–frequency characteristics and z, but also significantly on frequency band division. Traditional logarithmic segmentation offers generality but often causes redundant frequency points or suboptimal distributions in scenarios with uneven energy concentration or significant response variations, compromising computational efficiency and increasing sweep measurement burden. The sweep range must first be defined. Cumulative spectral energy distribution across frequencies is calculated using (25). Figure 6 displays the energy distribution of the 8/20 μs impulse (green solid line) and the system’s amplitude–frequency response (magenta dashed line). It can be observed that the signal energy is primarily concentrated between 5 kHz and 30 kHz. The cumulative energy reaches approximately 99% at 50 kHz and 99.9% at 100 kHz, indicating that the contribution of high-frequency components is negligible. The amplitude–frequency response remains relatively flat in the mid-band but shows significant attenuation at both ends, which may affect the synthesis accuracy of the impulse scale factor.

Due to the inverse relationship between the AC scale factor and the amplitude–frequency characteristics, coarse frequency division in the low-frequency band would amplify synthesis errors (hence the traditional dense point selection is maintained from 10 Hz to 100 Hz). Conversely, in the high-frequency band, fine division has minimal impact due to the low energy content (thus the traditional dense point selection is also applied from 100 kHz to 500 kHz). For the mid-frequency band, which contains the main energy of the impulse and exhibits a relatively flat system response, fewer frequency-sweep points are sufficient. Therefore, an adaptive division method based on spectral energy density and response characteristics is implemented to ensure accuracy while minimizing the number of sweep points. Specifically, as shown in the zoomed-in view of Figure 6, within the 5 kHz–29 kHz range (which accounts for approximately 80% of the total energy), the energy distribution increases almost linearly with frequency. Given the minimal variation in amplitude–frequency characteristics across this band, a linear frequency point selection approach can be adopted. This determines four frequency points—5 kHz, 13 kHz, 21 kHz, and 29 kHz—effectively reducing the sweeping workload while accounting for subtle differences in the amplitude–frequency response. Similarly, this method can be applied to select 1 kHz between 500 Hz and 5 kHz, and 36 kHz, 43 kHz, and 50 kHz between 29 kHz and 100 kHz.

Naturally, increasing the density of frequency points would improve accuracy but at the cost of additional sweeping effort. This proposed strategy covers the primary energy frequency bands while eliminating redundancy, thereby optimizing overall measurement efficiency.

$$\begin{array}{c}{S}_X\left(w\right)={\displaystyle \frac{\frac{1}{\mathsf{\pi }}{\displaystyle {\int }_0^w|X(jw){|}^2}dw}{\frac{1}{\mathsf{\pi }}{\displaystyle {\int }_0^{\infty }|X(jw){|}^2}dw}}\hfill \\ \hspace{1em}\hspace{1em} =-{\displaystyle \frac{4{\alpha }^3+4\alpha {\beta }^2}{\mathsf{\pi }}}{\displaystyle {\int }_0^w\frac{1}{{\left({\alpha }^2+{\beta }^2-w^2\right)}^2+4{\alpha }^2{w}^2}}dw\end{array}$$

(25)

To systematically evaluate the impact of model optimization and frequency band division strategies on synthesis accuracy, case studies were conducted using both band-pass and low-pass filters, respectively.

When the transfer characteristic of the measurement system follows a band-pass filter, as shown in

Table 2. Comparison of synthetic impulse scale factor accuracy using two division strategies (a band-pass filter).

Wl (Hz)	wu (MHz)	λ	Actual Impulse Scale Factor	Relative Error of Model Before Optimization (%) [19]	Relative Error of Optimized Model (Conventional Band Division)	Relative Error of Optimized Model (Proposed Band Division)
100	0.2	2000	370.8542	−3.4939	0.2047	0.1004
100	0.5	5000	366.1378	−5.3509	0.0929	0.0287
100	1	10,000	365.577	−5.6626	0.0483	−0.0176
10	0.2	2000	368.9269	2.3409	0.1367	0.1019
10	0.5	5000	363.9232	0.3636	0.0159	0.0185
10	1	10,000	363.2452	0.0291	−0.0296	−0.0291
1	0.2	2000	368.7338	2.4267	0.1805	0.1469
1	0.5	5000	363.7013	0.4483	0.0587	0.0623
1	1	10,000	363.0117	0.1136	0.0131	0.0146

, the relative error between the impulse scale factor obtained using the optimized model and the true impulse scale factor generally decreases as w_u increases or w_l decreases, regardless of whether the conventional or proposed frequency division method is applied. This trend confirms the necessity of broadening the measurement system’s bandwidth. In contrast, the unoptimized model exhibits larger errors and lacks a consistent pattern, reflecting poorer accuracy and stability, which further underscores the advantage of the optimized model. Notably, the proposed frequency band division method overall outperforms the conventional approach.

When the measurement system behaves as a low-pass filter, the results in

Table 3. Comparison of synthetic impulse scale factor accuracy using two division strategies (a low-pass filter).

wl (Hz)	wu (MHz)	λ	Actual Impulse Scale Factor	Relative Error of Model Before Optimization (%) [19]	Relative Error of Optimized Model (Conventional Band Division)	Relative Error of Optimized Model (Proposed Band Division)
0	0.2	3500	360.9538	0.9033	0.1124	0.1140
0	0.5	8750	359.3379	0.1595	0.0260	0.0279
0	1	17,500	359.1132	0.0415	0.0074	0.0079

show that as w_u increases, the relative errors of all three methods decrease significantly, with the performance improvement of the optimized model being particularly pronounced. However, for the same optimized model, the proposed frequency band division strategy performs slightly worse than the conventional method, which can be attributed to the specific amplitude–frequency characteristics of this type of system.

In practice, band-pass and low-pass filters correspond to measurement devices such as Rogowski coils and pulse shunts, respectively. The results above demonstrate that the proposed model optimization method and frequency division strategy exhibit good adaptability across different measurement system types.

Building upon the preceding analysis, a systematic comparison between the proposed and traditional models across multiple dimensions—including fractional-order characteristics, accuracy, and waveform applicability—is essential to visually demonstrate the refinements and advantages of our work. As demonstrated in

Table 4. Performance comparison of the novel model against the existing model.

Performance	Existing Model [19]	Proposed Model
Fractional-Order Characteristics	Unoptimized (default z = −1)	Optimization (adaptive determination of z-value based on time parameters of input and output waveforms)
Accuracy	Fair or poor	Higher
Applicable Waveforms	Only applicable to double-exponential waves (1.2/50 μs)	Suitable for double-exponential waves (1.2/50 μs) and exponentially damped sinusoidal oscillations waves (8/20 μs)

, the proposed approach not only improves accuracy but also significantly enhances the model’s applicability to different impulse waveforms.

5. Conclusions

A comparative analysis of the relative error in μ for different fractional-order parameter z values indicates that the model achieves optimal performance at z = 0.1, where the error between the synthesized impulse scale factor and the true value is minimized, and the model accuracy reaches its peak. To ensure the model’s validity for 8/20 μs waveforms, the deviation of the output waveform parameters T₁ and T₂ from the input waveform reference values should be controlled within 0.2%. Furthermore, for practical implementation, the frequency sweep range should extend to at least 200 kHz, and considering the signal-to-noise ratio, the sensitivity of the measurement device should not be lower than 1 mV/A.

Through a joint analysis of the system’s amplitude–frequency characteristics and the spectral energy distribution of the impulse, an optimized frequency band division strategy is proposed. This method maintains accuracy comparable to that of the conventional strategy while reducing the number of required frequency points by approximately 39%, thereby decreasing measurement workload and improving synthesis efficiency and operational convenience.

Compared to traditional models, the proposed model exhibits broader applicability and higher precision. It addresses a gap in the traceability of the impulse scale factor for impulse current measurements, significantly expands the applicable range of impulse waveforms, and strongly supports the refinement of the traceability theory for impulse scale factors.