Numerical Investigations into the Homogenization Effect of Nonlinear Composite Materials on the Pulsed Electric Field

Abstract

Pulsed power equipment is often characterized by high energy density and field intensity. In the presence of strong electric field intensity, charge accumulation within insulators exacerbates electric field non-uniformity, leading to potential insulation breakdown, thereby posing a significant threat to the safe operation of pulsed power equipment. In this manuscript, we introduce nonlinear composite materials with field-dependent conductivity and permittivity to adaptively regulate the distribution of the pulsed electric field in insulation equipment. Finite-element modeling and analysis of the needle-plate electrodes and high-voltage bushing are carried out to comprehensively investigate the non-uniformity of the distribution of the electric field and the homogenization effect of various nonlinear materials in the presence of pulsed excitations of different timescales. Numerical results indicate that the involvement of nonlinear composite materials significantly improves the electric field distribution under pulse excitations. In addition, variations in the rising time of the pulses affect the maximum electric field intensity within the insulators considerably, but for pulses of nanosecond and microsecond scales, the tendencies are the opposite. Finally, via the simulations of the bushing, we illustrate that some measures proposed for improving the uniformity of the electric field under low frequencies, e.g., increasing the length of the electric field equalization layer and the distance of the underside of the electric field equalization layer from the grounding screen, are still effective for the homogenization of pulsed electric field.

1. Introduction

With the rapid development of pulsed power technology, its applications continue to expand into fields including the environment, energy, defense, material processing, and medical treatment [1]. In general, pulsed power equipment is characterized by extremely high energy density and strong field intensity, which are likely to induce insulation breakdown, raising severe reliability and safety issues. Therefore, the distribution and homogenization of the electric field intensity under pulse excitation have received extensive attention.

In the presence of pulsed excitations, the distribution of the electric field intensity in insulators can be highly nonuniform [2,3], leading to potential surface flashover [4,5] and space charge accumulation [6,7,8,9,10,11,12]. The resultant insulation failure poses a severe threat to pulsed power equipment. Therefore, homogenization of the pulsed electric field, or in other words, to keep the spatial distribution of the electric field intensity uniform throughout the duration of the pulse, has been a major concern in pulsed power technology. Traditionally, the homogenization is achieved via improved design of the insulation structures [13,14]. Involvement of additional fan-shaped structures, equipotential rings, and optimization of the shapes of the spacers [15] or electrodes [16] are widely adopted measures to improve the uniformity of the electric field distribution. However, these approaches increase the difficulty of manufacture and maintenance of the equipment, and the effect of field homogenization is limited.

An alternative to the above measures is to employ composite materials with deliberately designed properties. In fact, research on material property modifications of composite materials to address flashover issues in insulation equipment has long been underway [17]. Initially, researchers employed composites with constant high electrical conductivity or permittivity to improve electric field distribution. In reference [18], the material parameters (permittivity, dielectric loss, surface resistivity, and volume resistivity) of insulation materials were extracted under different temperatures and frequencies via experiments, while in [19], an intuitive and accurate approach for modelling the electric conductivity of multiphase materials is proposed, making it possible to obtain and manipulate the effective material parameters of composites at very low costs. Compared to the traditional “shape control” methods, insulation by this “material-parameter control” method is simpler and more flexible. However, the homogenization effect by using composites with linear permittivity or electric conductivity are severely influenced by the temperature and frequency stabilities of the material parameters, and limited by the geometrical structure of the pulsed power equipment; thus, it is not completely satisfying. To overcome this bottleneck, some researchers involve nonlinear composites with field-dependent permittivity [20] or conductivity [21,22,23] rising with the increase in the electric field intensity. This adjustment achieves a match between the material parameters of insulation media and the local electric field intensity, thereby adaptively improving the uniformity of the electric field. Representative of such efforts is [24], in which a nonlinear conductive composite synthesized from a polymer matrix and inorganic fillers is proposed to improve the uniformity of electric field distribution. It is shown that nonlinear composite materials with field-dependent permittivity can adaptively homogenize the distribution of the electric field intensity, which is beneficial for insulation reliability.

It is worth noting that the available literature mainly focuses on the homogenization effect of merely nonlinear permittivity or conductivity under direct current (DC) condition or power frequencies. To the best of the authors’ knowledge, a comprehensive investigation on the joint effect of nonlinear electric conductivity and permittivity on the homogenization of the pulsed electric field is still lacking.

The contribution of this manuscript is twofold. First, finite element models for two typical scenarios, namely the needle-plate electrodes and the high-voltage bushing, are established. Through numerical results, the collaborative effect of nonlinear permittivity and electric conductivity on the homogenization of pulsed electric field intensity is illustrated. Second, based on the simulation results of the high-voltage bushing, we discuss the influence of the length and position of the grounding shield on the electric field distribution. Results indicate that measures to enhance electric field uniformity under low-frequency or DC conditions, such as increasing the length of the grounding shield, remain effective for improving uniformity under pulse excitations. The paper delves into the adaptive homogenization effect of field-dependent materials with nonlinear conductivity and permittivity, providing a theoretical basis for the design and optimization of the insulation of pulsed power equipment.

The remainder of this manuscript is structured as follows. In Section 2, the governing equations of the electromagnetic field in the insulator as well as the dependence of the nonlinear permittivity and conductivity are presented. A brief explanation on why field-dependent permittivity and electric conductivity are beneficial for yielding more uniform distribution of the electric field intensity is also included. In Section 3, finite element simulations for the needle-plate electrodes and the high-voltage bushing are carried out. The sensitivity analysis in terms of the mesh density and time-step size is first provided to validate the numerical results. Then, the impact of nonlinear material parameters, the time profiles of the pulse excitations, and geometrical parameters of the bushing on the electric field distribution are discussed in detail. The main conclusions of this research are summarized in Section 4.

2. Theory and Formulations

The electromagnetic field in the insulator in the presence of pulse excitations is governed by Maxwell’s equations:

$$\{\begin{array}{l}\nabla \times E=-{\displaystyle \frac{\partial B}{\partial t}}\\ \nabla \times H=J+{\displaystyle \frac{\partial D}{\partial t}}\\ \nabla \cdot D=\rho \\ \nabla \cdot B=0\end{array},$$

(1)

along with the constitutive law

$$\{\begin{array}{l}D=\epsilon E\\ B=\mu H\\ J=\sigma E\end{array},$$

(2)

where E is the electric field intensity, D the electric flux density, H the magnetic field intensity, B the magnetic flux density, J the electric current density, $\epsilon$ the permittivity, $\mu$ the permeability, and $\sigma$ the electric conductivity. As mentioned before, $\epsilon$ and $\sigma$ of nonlinear composite materials should be field-dependent:

$$\{\begin{array}{l}\sigma (\left|E\right|)={\sigma }_0\exp ({\alpha }_1\left|E\right|)\\ (\left|E\right|)={\epsilon }_0{\epsilon }_{r0}\exp ({\alpha }_2\left|E\right|)\end{array},$$

(3)

where $\left|E\right|$ is the absolute value of E, ${\sigma }_0$ the reference electric conductivity under $\left|E\right|$ = 0, ${\epsilon }_0$ the permittivity of the vacuum, and ${\epsilon }_{r0}$ the relative permittivity under $\left|E\right|$. The units of nonlinear coefficients ${\alpha }_1$ and ${\alpha }_2$ are both m/V, so that ${\alpha }_1\left|E\right|$ and ${\alpha }_2\left|E\right|$ are dimensionless.

It is worth emphasizing that the values of ${\alpha }_1$ and ${\alpha }_2$ are usually small positive numbers, such that when $\left|E\right|$ increases, $\epsilon (\left|E\right|)$ and $\sigma (\left|E\right|)$ rise accordingly. The permittivity $\epsilon$ is a measure of the materials’ capability to store energy and resist the variation in the electric field intensity via polarization. Therefore, in the presence of large $\left|E\right|$, enlarged $\epsilon (\left|E\right|)$ helps to weaken the local field to make the overall distribution of E more uniform. As for the electric conductivity, consider a small volume portion of the insulator. When local $\left|E\right|$ is large, $\sigma (\left|E\right|)$ defined by Equation (3) increases to reduce the effective resistance of this portion, such that the ratio of the overall voltage borne by this portion decreases, resulting in diminished local field and homogenization of the electric field intensity.

In summary, field-dependent $\epsilon (\left|E\right|)$ and $\sigma (\left|E\right|)$, rising with the increase in $\left|E\right|$, are both favorable for making the distribution of E more uniform. Though the nonlinear dependence of $\epsilon (\left|E\right|)$ and $\sigma (\left|E\right|)$ on $\left|E\right|$ can be fitted with many functions, including the linear function and high-order polynomials, in our context, exponential dependence of $\epsilon (\left|E\right|)$ [25] and $\sigma (\left|E\right|)$ [26] is assumed. By varying the coefficients ${\alpha }_1$ and ${\alpha }_2$, the strength of nonlinearity in terms of $\epsilon$ and $\sigma$ of the investigated materials can be easily adjusted.

3. Numerical Simulations of the Homogenization Effect of Nonlinear Composite Materials

This section carries out finite element (FEM) simulations in the presence of pulse excitations of nanosecond and microsecond scales for two models, i.e., needle-plate electrodes and high-voltage bushing. The FEM discretization of the governing equations is realized with the commercial FEM solver COMSOL Multiphysics version 6.0, and we choose the built-in backward difference method for time-domain solution to the governing equations.

The discussion in this section will focus on the homogenization effect of nonlinear composite material coefficients on the pulsed electric field inside insulators.

3.1. Configurations of the FEM Simulations

For convenience, we start with some parameters and definitions shared by both simulations.

In both simulations, the excitations are pulse voltages of different timescales imposed on the metallic electrodes. According to the IEC 61000-2-9 standard, the reference time profile of the nanosecond pulse voltage is

$${\mathrm{V}}_{\mathrm{ns}}(\mathrm{t})=26\times ({\mathrm{e}}^{-4\times {10}^7\mathrm{t}}-{\mathrm{e}}^{-6\times {10}^8\mathrm{t}}),$$

(4)

which corresponds to a front time t_pr = 5 ns, a peak value of 20 kV, and a half-width of 22 ns. In Equation (4), the units of time t and nanosecond pulse voltage ${\mathrm{V}}_{\mathrm{ns}}(\mathrm{t})$ are second (s) and kilovolt (kV), respectively. Similarly, according to the GB/T 3048.13-2007 standard, the reference time profile of the microsecond pulse voltage is set to be a lightning surge waveform given by

$${\mathrm{V}}_{\mathsf{\mu }\mathrm{s}}(\mathrm{t})=20.76\times ({\mathrm{e}}^{-1.4\times {10}^4\mathrm{t}}-{\mathrm{e}}^{-2.7\times {10}^6\mathrm{t}}),$$

(5)

which corresponds to a pulse front time of t_pr = 5 ns, a peak value of 20 kV, and a half-width of 50 ns. In Equation (5), the units of time t and microsecond pulse voltage V_μs(t) are second (s) and kilovolt (kV), respectively. Note that by adjusting the coefficients in Equations (4) and (5), it is straightforward to yield pulse waveforms of various front times t_pr.

The key indicator observed in the two simulations is the electric field non-uniformity, which is the ratio of the maximum magnitude of the field intensity E_max to the spatially averaged magnitude of the field intensity over the entire region E_av, denoted by P:

$$\mathrm{P}={\mathrm{E}}_{\max }/{\mathrm{E}}_{\mathrm{av}}.$$

(6)

Note that the non-uniformity P defined by Equation (6) is a dimensionless ratio.

The parameters of the linear materials involved in our simulations are listed in

Table 1. Parameters of linear materials.

Linear Material	$\mathit{\sigma }$ (S/m)	${\mathit{\epsilon }}_{\mathit{r}}$ (1)
Silicone rubber	1.00 × 10−14	4.3
Copper	5.80 × 106	1.0
Air	0	1.0

. Three types of nonlinear composite materials, referred to as S1, S2, and S3, respectively, are defined in

Table 2. Parameters of nonlinear composite materials.

Nonlinear Material	${\mathit{\epsilon }}_{\mathit{r}0}$ (1)	${\mathit{\sigma }}_0$ (S/m)	${\mathit{\alpha }}_1,{\mathit{\alpha }}_2$ (m/V)
S1	4.3	10−14	10−5	10−7
S2			2 × 10−5	2 × 10−7
S3			3 × 10−5	3 × 10−7

. The three nonlinear composite materials mainly differ in the strength of nonlinearity, or more specifically, S1 has the weakest nonlinearity, while S3 exhibits the strongest nonlinearity.

Figure 1 and Figure 2, respectively, present the nonlinear conductivity curves and nonlinear permittivity curves of the three composite materials. Note that the “1” in the bracket indicates that the relative permittivity is a dimensionless quantity. It is worth noting that though the investigated nonlinear materials S1, S2, and S3 with nonlinear properties do not correspond to any real-world material, the magnitudes are close to the reported values in [26] or determined by fitting Equation (3) using the experimental data in [27].

3.2. Needle-Plate Electrode Model

The needle-plate electrode model is a typical structure with an extremely uneven electric field distribution. As shown in Figure 3, a cylindrical sample with a thickness of 40 mm and a diameter of 20 mm [20,28] is placed between the needle and plate electrodes. To ensure good contact between the electrodes and the sample, the tip of the needle electrode is designed with a flat end of 10 mm width, with radii of 5 mm on both sides. Reference line 1 is set along the axis of the sample, and reference line 2 is placed along the bottom surface of the sample. Pulse excitations are applied at the tip electrode, while the plate electrode is grounded. The electrodes are made of copper.

Figure 4 illustrates the axial electric field distribution of the needle-plate electrode configuration with nonlinear composite material insulators under pulse voltage of microsecond scale, observed at the time instant when the pulse reaches its peak. It is evident that the electric field intensity is primarily concentrated near the tip electrode.

For validation of the numerical results, sensitivity tests of the FEM model of the needle-plate electrodes are conducted to exclude the interference of mesh density and time-step size.

In terms of mesh density, when compared with the pulsed field of the microsecond scale, phenomena of the nanosecond scale require a denser grid; thus, only the sensitivity test in the presence of the nanosecond pulse needs to be carried out, and the same mesh can be trusted for the pulsed field of the microsecond scale.

Table 3. Sensitivity test of the needle-plate model in terms of mesh density. The excitation is the nanoscale pulse with t_pr = 5 ns. The investigated nonlinear material is S2 defined in Table 2.

Number of Elements	Maximum Edge Length (mm)	Emax (kV/mm)	P
3468	10.51	6.246	15.985
5172	7.04	6.276	14.614
9210	3.89	6.253	14.537
17,725	2.10	6.265	14.211
61,377	1.05	6.288	14.219

presents the results of E_max and P excited by a nanosecond-scale pulse of t_pr = 5 ns under various numbers of tetrahedron elements. It can be seen that as the number of elements exceeds 9210, E_max and P both converge to their respective grid-independent values. For balance between accuracy and efficiency, the number of mesh elements is set to 17,725 in our context.

When it comes to the determination of the time-step size, we conduct sensitivity tests under both nanosecond and microsecond scales for different time-stepping methods, as shown in

Table 4. Sensitivity test of the needle-plate model in terms of time-step size. The excitation is the nanosecond-scale pulse with t_pr = 5 ns. The investigated nonlinear material is S2 defined in Table 2.

Time-Stepping Method	Maximum Time-Step Size (ns)	Time Iterations	Emax (kV/mm)	P
Backward difference	2	326	6.265	14.211
	0.1	1161	6.275	14.608
	0.01	10,170	6.263	14.41
Generalized alpha	0.01	10,022	6.268	14.482
	0.001	100,011	6.258	14.373

and

Table 5. Sensitivity test of the needle-plate model in terms of time-step size. The excitation is the microsecond-scale pulse with t_pr = 10 μs. The investigated nonlinear material is S2 defined in Table 2.

Time-Stepping Method	Maximum Time-Step Size (μs)	Time Iterations	Emax (kV/mm)	P
Backward difference	2	128	8.363	29.270
	0.1	1114	8.367	29.365
	0.01	10,017	8.365	29.175
Generalized alpha	0.01	10,022	8.363	29.168
	0.001	100,009	8.353	29.099

. It is worth noting that the two built-in time-domain solvers, i.e., the backward difference method and generalized Alpha method, are both implemented with adaptive time-step size. In each time advance, these solvers conduct a posterior error estimation and adjust the time-step size accordingly. Therefore, the time-step sizes are not fixed, but we can control the maximum time-step sizes. It can be seen that the results via the two time-stepping methods agree very well. For nanosecond-scale analysis, a maximum time-step size of 2 ns combined with the backward difference solver is sufficient regarding accuracy, while for microsecond-scale simulations, the maximum time-step size is chosen to be 2 μs.

To demonstrate the homogenization effect of the nonlinear composite materials, the evolution of the electric field non-uniformity P under pulse excitations of microsecond and nanosecond scales is depicted in Figure 5a,b, respectively. In Figure 5 and Figure 6, the considered nonlinear permittivity $\epsilon (\left|E\right|)$ and electric conductivity $\sigma (\left|E\right|)$ are

$$\{\begin{array}{l}\epsilon (\left|E\right|)=4.3{\epsilon }_0\exp ({10}^{-7}\left|E\right|)\\ \sigma (\left|E\right|)={10}^{-14}\exp ({10}^{-5}\left|E\right|)\end{array}.$$

(7)

Note that the units of $\epsilon (\left|E\right|)$ and $\sigma (\left|E\right|)$ are F/m and S/m, respectively. When nonlinear $\epsilon (\left|E\right|)$ and $\sigma (\left|E\right|)$ are both involved, the considered material is S1, defined in Table 2.

It can be seen in Figure 5a that under pulse excitations of microsecond scale, non-uniformities P in the silicone rubber sample are almost invariant. Involvement of nonlinear ${\epsilon }_r$ can homogenize the field distribution a little, but at the time instants when the pulse voltages reach their peaks, the improvement of non-uniformities P is approximately 10%, which is not significant. For comparison, field-dependent $\sigma$ exhibits an enhanced homogenization effect, but the loss is also increased due to the rising $\sigma$. When the sample material has both nonlinear ${\epsilon }_r$ and $\sigma$, the distribution of E can be further optimized. During the pulse front time (t < 20 μs), P can be reduced by 25%, demonstrating the benefit of nonlinearity in both ${\epsilon }_r$ and $\sigma$. Similar conclusions can be drawn from Figure 5b.

Figure 5. Time evolution of P (non-uniformity of the distribution of E) under pulsed excitations. (a) Microsecond pulse. (b) Nanosecond pulse.

Figure 5. Time evolution of P (non-uniformity of the distribution of E) under pulsed excitations. (a) Microsecond pulse. (b) Nanosecond pulse.

The results in Figure 6 indicate that under pulse excitations of nanosecond scale, the influence of the pulse front time t_pr is crucial. It can be seen that the proposed material with nonlinear ${\epsilon }_r$ and $\sigma$ can still homogenize the distribution of E at the beginning of the duration. However, for t_pr = 20 ns, it can be seen that P of the proposed material can even exceed that of the silicone rubber sample. Nevertheless, this does not mean that the joint effect of nonlinear ${\epsilon }_r$ and $\sigma$ is unfavorable. In Figure 7, the maximum electric field intensities ${\mathrm{E}}_{\max }$ in the presence of nanosecond excitations corresponding to various t_pr are presented. It is obvious that the sample material with nonlinearity in both ${\epsilon }_r$ and $\sigma$ exhibits the smallest E_max, leading to minimum risk of insulation breakdown as well as lower loss when compared against the sample with merely nonlinear $\sigma$.

Figure 6. Time evolution of P (non-uniformity of the distribution of E) under excitations of various t_pr. (a) Microsecond pulse. (b) Nanosecond pulse.

Figure 6. Time evolution of P (non-uniformity of the distribution of E) under excitations of various t_pr. (a) Microsecond pulse. (b) Nanosecond pulse.

Another point is that the silicone rubber, as a linear material with constant permittivity and electric conductivity, exhibits almost invariant behavior under both timescales and various t_pr, because the distribution of the internal physical field is determined mostly by its material properties.

On the other hand, t_pr has much more obvious influences on the behavior of the proposed nonlinear materials. A possible explanation is that the nonlinear variations in the material properties are in fact a reflection of the charge relaxation process [27]; thus, t_pr and the resultant time profiles of pulsed excitations have significant influence on the non-uniformity P of the insulators made of nonlinear materials with field-dependent permittivity and electrical conductivity, especially when the timescale of the pulsed field is comparative or shorter than the time-constant of the relaxation process. From Figure 6, it can be seen that the influence of t_pr is more obvious for nanosecond-scale pulses, which is consistent with the above discussion.

Then, we discuss the influence of the strength of nonlinearity on the homogenization effect of sample materials.

Table 6. Numerical results of insulator samples made of materials of different strengths of nonlinearity.

Excitations	Sample Material	Maximum Field Intensity Emax (kV/mm)	Electric Field Non-Uniformity (P)
		tpr = 5, 10, 20 ns	tpr = 5, 10, 20 ns
Nanosecond pulse	Silicone rubber	13.122, 13.126, 13.140	32.490, 32.490, 32.490
	S1	9.103, 4.664, 8.180	16.760, 12.400, 29.555
	S2	6.265, 4.385, 3.293	14.211, 10.257, 7.236
	S3	3.228, 3.342, 2.463	7.203, 7.707, 5.611
		tpr = 2, 10, 20 μs	tpr = 2, 10, 20 μs
Microsecond pulse	Silicone rubber	13.200, 13.158, 13.100	38.888, 38.590, 38.871
	S1	9.813, 9.986, 10.092	38.497, 37.530, 37.624
	S2	8.046, 8.363, 8.853	32.143, 29.270, 28.986
	S3	7.191, 7.544, 8.139	27.731, 25.267, 29.250

provides the maximum electric field intensity and field non-uniformity along axis 1. For a more intuitive summarization of the simulation results, we also include Figure 8 to present Emax corresponding to various t_pr under excitations of both timescales.

It can be concluded that for the needle-plate model, whether excited by microsecond or nanosecond pulses, stronger nonlinearity, or in other words, larger nonlinear coefficients ${\alpha }_1$ and ${\alpha }_2$, can help to achieve better field uniformity. Additionally, as the pulse front time t_pr increases, the maximum electric field intensity E_max within the sample excited by microsecond pulses gradually increases, whereas for excitations of nanosecond scale, roughly the opposite tendency can be observed.

Figure 9 and Figure 10 depict the temporal variation in electric field non-uniformity P inside the sample under pulse excitations of nanosecond and microsecond scales. It can be concluded that throughout the pulse rising stage of the ns pulse, as the nonlinear coefficients increase, P gradually decreases, which is favorable. However, during the falling stage corresponding to the tails of the waveforms, an unignorable increase in the electric field non-uniformity P appears for sample materials with stronger nonlinearity. However, the pulse excitations as well as the maximum electric field intensity have decayed to small values; hence, the insulating performance of the nonlinear materials may not be affected. As for the cases with pulse excitations of microsecond scale, throughout the rising stage of the μs pulses, the electric field non-uniformities P of nonlinear samples made of S1, S2, and S3 remain consistently lower than that of the silicone rubber sample.

3.3. High-Voltage Bushing Model

In this subsection, the distribution of electric field intensity in a high-voltage bushing [20] is investigated. The geometry of the bushing is shown in Figure 11, where the reference values for H and are H = 1000 mm and d = 5 mm, respectively. The structure of the bushing consists of the flange, guide bar, insulation layer, grounding screen, and an electric field equalization layer made of silicone rubber or nonlinear composite materials. The electric field distribution in the insulation layer is plotted in Figure 12, at the time instant when the pulse excitation of microsecond scale reaches its peak. It can be observed that the electric field intensity is mainly concentrated between the grounding shield and the guide bar.

The key indicators of this simulation are still the non-uniformity P and maximum amplitude of the electric field intensity E_max in the bushing. Similar to the process in Section 3.1, sensitivity tests of the FEM model of the bushing are first carried out to validate the simulation results. From the results in

Table 7. Sensitivity test of the high-voltage bushing model in terms of mesh density. The excitation is the nanoscale pulse with t_pr = 5 ns. The investigated nonlinear material is S2, defined in Table 2.

Number of Elements	Maximum Edge Length (mm)	Emax (kV/mm)	P
6087	711	1.482	11.75
8298	216	1.485	12.15
11,234	144	1.483	11.18
36,613	21.6	1.482	11.25

,

Table 8. Sensitivity test of the needle-plate model in terms of time-step size. The excitation is the nanosecond-scale pulse with t_pr = 5 ns. The investigated nonlinear material is S2, defined in Table 2.

Time-Stepping Method	Maximum Time-Step Size (ns)	Time Iterations	Emax (kV/mm)	P
Backward difference	2	105	1.483	11.18
	0.1	1023	1.485	11.23
	0.01	10,017	1.487	11.20
Generalized alpha	0.1	1125	1.485	12.06
	0.01	10,007	1.481	11.16

and

Table 9. Sensitivity test of the needle-plate model in terms of time-step size. The excitation is the microsecond-scale pulse with t_pr = 10 μs. The investigated nonlinear material is S2, defined in Table 2.

Time-Stepping Method	Maximum Time-Step Size (μs)	Time Iterations	Emax (kV/mm)	P
Backward difference	2	153	1.261	9.463
	0.1	1041	1.265	9.451
	0.01	10,017	1.265	9.445
Generalized alpha	0.01	10,022	8.363	9.426
	0.001	100,009	8.353	9.351

, the number of mesh elements is set to be 11,234 in the following discussions, and the maximum time-step sizes in the presence of nanosecond and microsecond pulses are 2 ns and 2 μs, respectively. The adopted time-stepping method is the backward difference method.

Then, we consider the equalization layer made of silicone rubber and nonlinear materials S1, S2, and S3. Figure 13 presents the maximum electric field intensity E_max in the high-voltage bushing corresponding to various t_pr and excitations of both timescales. It is found that regardless of the timescales of the pulse excitations, the introduction of nonlinear materials is still effective for reducing E_max, and enhancing the strength of nonlinearity improves the homogenization of the distribution of electric field intensity.

Figure 14 and Figure 15 depict the temporal variation in the electric field non-uniformity P in response to nanosecond and microsecond pulses, respectively. It can be observed that P can be significantly reduced by involving nonlinear materials, and stronger nonlinearity is generally beneficial to yield more uniform distribution of the electric field intensity. It is notable that during the latter half of the simulated duration, P seems to increase, and in some cases, P can even exceed those corresponding to the linear material, i.e., silicone rubber. However, at the end of the simulated duration, the excitations as well as E_max have decayed to very low levels, roughly 20% of the peak values; thus, the risk of insulation failure is limited.

The above conclusions are similar to those obtained in Section 3.2. However, for this model, we extend our discussion to investigate whether some common approaches for electric field homogenization under power frequency or direct current (DC) conditions are still effective for pulse excitations of microsecond and nanosecond scales. In [26], it is demonstrated that the variation in length and position of the electric field equalization layer in bushings also affects the maximum electric field intensity and field non-uniformity within the insulation layer. S2, defined in Table 2, is chosen as the nonlinear material for the electric field equalization layer. Define three possible lengths of the layer H1= 1000 mm, H2 = 1300 mm, and H3 = 1600 mm, as well as three possible distances of the grounding shield from the bottom of the layer, i.e., d1 = 0 mm, d2 = 5 mm, and d3 = 10 mm.

Table 10. Numerical results of the high-voltage bushing with electric field equalization layers of various lengths and positions.

Excitations	Length and Position of the Layer		Maximum Field Intensity Emax (kV/mm)	Electric Field Non-Uniformity (P)
			tpr = 5, 10, 20 ns	tpr = 5, 10, 20 ns
Nanosecond pulse	H1(1000 mm)	d1(0 mm)	2.14,2.57,2.11	16.17,19.33,15.89
		d3(10 mm)	1.89,1.85,1.84	14.24,13.93,13.87
		d2(5 mm)	1.57,1.48,1.59	11.91,11.24,11.96
	H2(1300 mm)		1.60,1.54,1.60	12.09,11.61,12.02
	H3(1600 mm)		1.65,1.80,1.67	12.52,13.52,12.60
			tpr = 2, 10, 20 μs	tpr = 2, 10, 20 μs
Microsecond pulse	H1(1000 mm)	d1(0 mm)	1.88,1.75,1.82	14.13,13.29,13.75
		d3(10 mm)	1.57,1.48,1.48	11.78,11.19,11.09
		d2(5 mm)	1.40,1.26,1.19	10.47,9.46,8.96
	H2(1300 mm)		1.26,1.21,1.16	9.40,9.11,8.76
	H3(1600 mm)		1.25,1.18,1.13	9.37,8.90,8.51

presents detailed simulation results regarding the impact of varying lengths and positions of the electric field equalization layer on the maximum electric field intensity E_max and field non-uniformity P within the insulation layer under various excitations.

Figure 16 intuitively summarizes the variation in E_max under fixed H = 1000 mm while varying positions and t_pr. It can be seen that there seems to be an optimal position for the grounding shield, since the curves are not monotonous, which agrees well with the conclusions for DC and power frequency scenarios in [26]. Denser sampling of the parameter d may help to determine the optimal position of the grounding shield. In addition, the influences of d under both microsecond and nanosecond excitations are almost identical; thus, unified experience can be applied to the design of bushing under both timescales.

Figure 17 depicts the variation in E_max under fixed d = 5 mm while varying H and t_pr. It is obvious that under excitations of microsecond scale, increase in the length H of the equalization layer can reduce E_max, which is also consistent with the results in [26]. However, given excitations of nanosecond scale, the experience is not valid anymore. Exactly the opposite tendency is observed. Therefore, determination of the length H of the equalization layer in the presence of high-frequency excitations requires deliberate investigation.

In summary, previous experience for homogenization of the electric field intensity under DC or power frequencies is still trustworthy in the presence of pulse excitations of microsecond scale. Nevertheless, faced with pulse excitations of nanosecond scale, some abnormal phenomena appear; thus, measures for homogenizing DC or power frequency electric field cannot be directly inherited. Deliberate validation of existing homogenization approaches is required prior to applications to nanosecond-scale scenarios.

4. Conclusions

In this manuscript, the homogenization effect of nonlinear composite materials with field-dependent permittivity $\epsilon$ and conductivity $\sigma$ on the distribution of pulsed electric field intensity is thoroughly investigated numerically. Two key conclusions are obtained. The findings and limitations of this manuscript are briefly summarized in

Table 11. Main conclusions of this manuscript.

	Conclusions	Explanations
Findings	♦ The composite materials with nonlinear $\epsilon$ and $\sigma$ exhibit the best homogenization effect.	♦ Field-dependent $\epsilon$ and $\sigma$ can adaptively regulate the local electric field.
	♦ The time profile and tpr of the pulses have considerable influence on the behavior of the nonlinear materials.	♦ Due to the charge relaxation process reflected by the nonlinear parameters, tpr and time profiles of the pulses have great impact on the behavior of the nonlinear materials, especially under nanosecond scales.
	♦ Homogenization measures developed for DC and power frequency condition are still trustworthy for the microsecond scale, but may not be valid for the nanosecond scale.	♦ The nonlinear materials exhibit abnormal behavior under nanosecond scales due to the relaxation process, which is similar to the case in the second finding.
Limitations	♦ Experimental validation is absent in the current research.	♦ Lack of electric field sensors free of metallic constituent. ♦ Deployment of the sensors can interfere the considered electric field.

, and detailed interpretations are as follows.

• The nonlinear composite materials with field-dependent $\epsilon$ and $\sigma$ exhibit a significantly improved homogenization effect compared to those of the linear materials, e.g., silicone rubber, and materials with merely nonlinear $\epsilon$ or $\sigma$. In addition, enhancement of the nonlinearity of the materials is generally favorable for field homogenization.
• The distribution of the internal physical field of the linear material, i.e., silicone rubber, is determined mostly by its material properties, and the non-uniformity P changes very little under various t_pr. On the other hand, the nonlinear variations in the material properties are in fact a reflection of the charge relaxation process; thus, t_pr and the time profile of the pulsed excitations have significant influence on the non-uniformity P of the nonlinear materials, especially when the timescale of the pulsed field is comparative or shorter than the time-constant of the relaxation process. It can be seen that the influence of t_pr is more obvious for nanosecond-scale pulses, which is consistent with the above explanation.
• Existing experience for homogenization of the electric field intensity under DC or power frequencies is still valid in the presence of pulse excitations of microsecond scale. However, compared with the electric field generated by DC or power frequency sources, the pulsed electric field of nanosecond scale behaves in a different manner; thus, some abnormal phenomena appear. Therefore, inheritance of the measures commonly adopted in DC or power frequency scenarios demands cautious evaluation and implementation when applied to the homogenization of the pulsed electric field of nanosecond scale.

It is worth emphasizing that the limitation of the present research mainly lies in the absence of experimental validation. To evaluate the non-uniformity of the distribution of the electric field intensity within the insulators, measurement of the internal electric field is demanded, which is a troublesome task due to the lack of electric field sensors free of metallic constituents and deliberate deign of experimental apparatus to avoid the sensors interfering with the considered electric field. The development of a dedicated experimental platform and electric field sensors itself is a valuable research topic and will be our focus in the future.