Modeling of the Transient Electric Field in Multilayer Dielectric Composites Under Impulsive HV Energization

This article presents the theoretical analysis of composite electrical insulation, formed from layered dielectric materials and subjected to impulsive energization. The 1-D planar and cylindrical geometries were considered, consisting of an arbitrary number of layers with arbitrary relative permittivity and electrical conductivity. Analytical solutions have been successfully derived for the time-dependent electric field inside the ${i}$ th layer. To demonstrate the usage of the model under complex multilayer topologies where analytical solutions are nontrivial, the characteristics of a 20-layer-graded composite under microsecond and sub-microsecond impulses were analyzed and validated against a finite-element (FE) solver. Results indicate that the transient electric field response under impulsive energization is strongly dependent on the interplay between the composite relaxation time constants and the characteristic timescales associated with the applied impulse. The model is a further development for the design and coordination of functionally graded materials (FGMs) and composite insulation for high-voltage (HV) system design. This is particularly relevant under fast-rising impulsive conditions as often encountered in many pulsed power applications.


I. INTRODUCTION
I N HIGH-voltage (HV) power and pulsed-power equipment, the presence of fast-rising impulsive voltages can impose a significant degree of transient electrical stress on system components, which often presents a unique and difficult challenge for insulation design. Failure to account for peak field enhancement at critical regions, such as triple junctions, may lead to insulation failure and electrical breakdown. In recent years, there has been increased research interest in novel field grading methods, motivated by the ever-increasing complexity of HV systems, the drive toward system miniaturization, and the development of novel pulsed power technology. Examples include various geometrical modifications to existing system accessories, the use of surface grading through conductive coatings [1], [2], or the use of field-adaptive grading materials [3], [4], [5], [6]. Promising developments are functionally graded materials (FGMs) and composite materials, which employ spatially nonuniform distributions of permittivity and/or electrical conductivity as a means of field grading. It is remarked that the conventional field-grading materials designed with field-adaptive characteristics may also be referred to as FGMs, e.g., [3], [4], [5], [6], but the focus of this study is on materials with spatially nonuniform properties. Both continuous [7] and layered FGMs [8] have been described in the literature, where the former typically provides a smoother field distribution, but with significant manufacturing complexity and at higher cost when compared with the latter. However, in recent times, progress has been made toward more effective fabrication techniques for FGMs, e.g., centrifugal methods using particulate fillers [9], fused-deposition modeling (FDM) [10], the flexible mixture casting (FMC) method [11], or injection-based techniques [12], among several others. Indeed, each method has its own strengths, drawbacks, and optimization techniques, which will not be described in detail within this study. For this, the reader is referred to [9], [10], [11], [12], [13], and references therein.
A number of published studies have successfully demonstrated the application of FGMs to HV systems, with the majority focused on gas-insulated system (GIS) spacers for ac or HVdc power transmission applications. An earlier paper by Hayashi et al. [14] reported on the breakdown behavior of layered composites with a relative permittivity range of ε r = 3.0-5.9 under ac and lightning impulses, comparing their dielectric performance to those of homogeneous samples with ε r = 5.9. Experimental results revealed around 10%-25% increase in the breakdown voltage associated with the flashover of graded spacers under positive lightning impulses. The improvement observed from the layered FGM can be explained by the field grading effect, i.e., redistribution of the field across the gas-spacer interface. Using a disk-type spacer topology between plane-plane electrodes, simulations presented in [14] showed 23% and 44% reduction in tangential field strength for spacers following inverse and inverse-square grading profiles, respectively. Kurimoto et al. [15] demonstrated FGMs with a "U"-shaped permittivity profile for radially graded cone-type GIS spacers. Modeled results indicated 38%-63% and 51%-62% field reduction near the anode and cathode triple junctions, respectively. The authors describe the fabrication of the spacer using an epoxy resin matrix with TiO 2 and Al 2 O 3 microparticles as fillers, and estimations presented in [15] suggest that the FGMs may also possess superior operational lifetime over uniform materials.
Other studies conducted during a similar period include work by Ju et al. [16], [17] who presented an optimization methodology for the design of GIS spacers using a novel elliptical grading profile, predicting ≈26% reduction of the maximum field strength. More recent studies by Hayakawa et al. [9], [18] present a manufacturing technique for radially graded GIS spacers using a centrifugal method. An analytical model was also developed and validated against experimental data, which was able to predict the resultant permittivity distribution of the FGM. In [11], a further casting method is shown for axially graded spacers using SiTiO 3 and SiO 2 filler microparticles in epoxy resin-the authors analytically predict up to 70% increase in discharge inception voltage in SF 6 at reduced pressures. Computational studies were subsequently conducted by Imai et al. [12], [19], which overall demonstrated that resin distributions resulting from the casting method could be reasonably predicted from simulation.
Du et al. [20] combined a traditional volume-graded FGM spacer with surface flourination, reporting up to 36% higher dc flashover strength. Zhang et al. [21] investigated the application of novel carbon nanotube-doped SiR material for power cable joints, where the multilayer and pointwise FGMs were studied under ac and dc energization and found that the FGM joints can substantially improve field uniformity. Talaat et al. [22] performed a computational study on FGM cone-type spacers using radially graded multilayer composites in the presence of a conducting particle. Their findings suggest that the application of certain FGMs may reduce the susceptibility of spacer interfaces to local field enhancement around contaminant particles. A detailed computational model was developed by Rachmawati et al. [7], who studied the influence of various permittivity and conductivity profiles on GIS spacers. In their work, the temperature and the field dependence of conductivity were also modeled under dc steady state, dc polarity reversal, dc-on (which incorporates a transition from zero to full applied voltage), and dc steady-state superimposed with lighting impulse conditions. The authors conclude that a "U"-shaped permittivity profile, combined with a conductivity distribution, which only increases near the ground electrode, was, overall, the most effective at reducing the maximum field at the triple junction.
Multilayer composites under impulsive regimes, however, have been studied to a far lesser extent. Under transient conditions, both the permittivity and electrical conductivity of each composite layer play a crucial role in determining the performance and characteristics of the insulating system. Modern pulsed power systems often utilize fast-rising impulses with rise times in the range of hundreds of picoseconds to microseconds [23] and have found applications across many areas of engineering. Some examples include high-speed switching technology [24], pulsed electric field (PEF) treatment for biomedical, and food processing industries [25], [26], to pollution control [27] and defense applications [28]. Therefore, there is an increased interest to characterize novel insulation solutions under fast transient signals and in nonstandard topologies, as it has become critical for the optimization and further development of these technologies. However, there exist few scenarios where fast transient stresses can be treated in the same way as steady state ac or dc, therefore necessitating a fully transient approach to the analysis.
In this study, a theoretical approach for analyzing the transient electric field inside multilayered composites is presented. An analytical model has been developed for the estimation of the electric field developed inside an n-layer, poorly conducting composite under fast-rising impulses. The motivation to employ an analytical approach is to have the ability to gain far deeper insights into the underlying physical behavior compared with computational modeling. It additionally allows for the rapid estimation of the relevant design parameters, far faster than the use of conventional finite-element (FE) analysis and at minimal computational cost, facilitating parametric and optimization studies. Using this model, the transient field characteristics have been analyzed, and the capability of the model is further demonstrated in the context of FGMs for GIS spacers.

A. Model Geometry and Governing Equations
A 1-D planar configuration was initially considered in this work, comprised of n layers of poorly conducting material (i.e., material that is characterized by its relative permittivity and electrical conductivity), stacked between parallel-plane electrodes. A cross section of this layered structure is shown in Fig. 1. Each layer indexed i is defined by a constant relative permittivity, ε i , electrical conductivity σ i in S/m, and thickness h = (z i − z i−1 ) in meters, where z i is the z-coordinate of the boundary between the i th and (i + 1)th layers. As the primary purpose of this study is for fast-rising pulsed conditions, the temperature dependence of conductivity was neglected, as there would be no significant change in temperature over such short timescales. It is also well known that the conductivity exhibits a highly nonlinear dependence upon the electric field, such that its inclusion would almost certainly render an analytical solution impossible. This dependence has, therefore, been omitted from this work, and constant conductivity is assumed. However, it should be noted that there exist two important cases for which this assumption is valid: 1) under sufficiently low field or 2) fast-rising voltage conditions. The first pertains to the far weaker dependence of σ (E) for common insulating materials below an approximate value of 10 6 -10 7 V/m (see, for instance, [29]). The second assumes that the conductivity requires time to rise, as investigated in some materials in [30] and [31], and that the relaxation time of the conduction process [30] is far longer than the rise time of the applied impulse. Under either one of these conditions, the conductivity can reasonably be assumed to exhibit minimal change with the field. It should further be noted that even if it were possible to incorporate σ (E) dependence into the model using the existing expressions found in the literature, these are typically derived under steady-state conditions, and their validity under fast transient conditions remains to be fully established. For boundary conditions, it was assumed that the stack would be energized through an arbitrary time-varying voltage U 0 (t) applied to the HV electrode, while the ground electrode is held at 0 V.
The potential field developed in the stack, assuming negligible space-charge effects, will be governed by Laplace's equation, which in the present 1-D case reads is the spatially and temporally varying electric potential in the i th layer with units of volts. Equation (1) has the general solution where A i (t) and B i (t) are time-dependent coefficients unique to each layer. The corresponding electric field E i (t) is computed from the gradient of the potential, from which one finds

B. Boundary Conditions
At the electrodes, the ground potential at z = 0 and applied potential U 0 (t) at z = z n must be prescribed. In addition, at each interface between two adjacent layers, the potential must be continuous across the boundary. These potential conditions take the form of the following equations: Furthermore, the current continuity condition of (7) must be applied across each interface, stipulating that the sum of the conduction and displacement currents at each side of a layer boundary must be equal. This condition additionally takes into account the accumulation of unbound surface charge due to the Maxwell-Wagner effect III. ANALYTICAL SOLUTIONS

A. Solution in the s-Domain
The system formed by (1)- (7) can be converted to simple algebraic forms through the application of the Laplace transform, from which one can show that, in the Laplace s-domain which takes the form of a recursive addition from i = 1 to i = n and can be expanded and rearranged to give Following (4), it can be shown that B 0 (s) = U 0 (s). As such, its substitution into (9) yields the expression Rearranging the s-domain form of (7) provides a similar recursive relation between the A coefficients of adjacent layers, ultimately allowing A i (s) to be expressed as follows: where τ i is the intrinsic relaxation time constant for the i th layer, defined as follows:

B. Existence of Closed-Form Time-Domain Solutions
To model the field distribution in different practical multilayered systems, it would be convenient to obtain a closedform time-domain solution for the field distribution. Analysis of the general s-domain solution (11) has been conducted, and one can show that there exists an upper limit to n for which a closed-form time-domain solution is possible. An expanded form of (11) can be written as follows: where the denominator can be shown to be a polynomial in s with degree n − 1. This is to say ≡ a 0 s n−1 + a 1 s n−2 + a 2 s n−3 + · · · + a n−2 s 1 + a n−1 s 0 (14) where the values of a 0 , . . . , a n are constant coefficients that are strictly negative. As such, the factorization of the denominator must result in (15) where the additional n − 1 time constants are derived from the roots of the scaled characteristic polynomial of the system and, hence, Considering (16) and (17), the Abel-Ruffini theorem [32] states that the roots to a polynomial of degree ≥5 with arbitrary coefficients cannot be expressed in radicals. Thus, one concludes that no closed-form solutions to (17) can be found for n > 5. For n = 1, (11) reduces to the elementary solution U 0 (s)/z 1 . For n = 2, the degree of P n is 1, thus allowing (17) to be solved by simple rearrangement. For n = 3-5, the roots can be found by means of the closed-form quadratic, cubic, and quartic formulas, respectively. If n exceeds 5, a solution can be sought by solving (17) via any preferred numerical root-finding method, such as the application of the eigenvalue theorem, which are available in popular symbolic math toolboxes. Examples with n = 20 are presented from Section IV onward. Up to this point, it should be noted that the analysis has been conducted, assuming an arbitrary s-domain voltage U 0 (s). The general solution (11) is, therefore, applicable to any applied waveform, so long as the signal has an analytical s-domain form. For instance, sinusoidal ac, dc, exponential impulses, ramps, or superpositions of the above are all equally possible.

C. Time-Domain Solution for Impulsive Voltage
In this work, time-dependent impulsive voltages were of interest, which are applied to the HV electrode. The voltage waveform is analytically represented by the commonly used double-exponential form of the following equation: where U 0 is the peak voltage magnitude in volts, and A 0 is a dimensionless scaling parameter, while α and β are wave-shaping parameters in s -1 , which control the rise (front) and fall (tail) characteristics of the impulse. Applying the Laplace transform to (18) and using (11) yield the s-domain double-exponential solution from which the inverse transform allows the derivation of the time-domain field magnitude inside any layer i where The coefficients given by (21) together with the roots from (16) show that the transient response of the electric field in any layer is determined by the complex interplay between the relaxation characteristics of every other layer present in the composite and of the transient characteristics of the applied impulse. This aspect is analyzed in further detail from Section IV onward.

D. Extension to the Cylindrical Domain
Cylindrical conductors with concentric insulation layers are of significant practical importance to power and pulsed power systems. Examples include cables with conductors wrapped in multiple layers of insulation, or GIS spacers, which may be graded in the radial direction. It can be shown that with minimal changes to (11), the model can be extended to a cylindrical domain.
Consider a cylinder of radius r , as shown in Fig. 2, from which we may approximate the geometry using 2-D polar coordinates. If the inner conductor of radius r 0 is wrapped in n layers of material, then the radius of the i th layer is denoted as r i . It is further assumed that the outer layer at r = r n is grounded. The general solution of the Laplace equation in the radially symmetric polar domain takes the form where C i (t) and D i (t) are once again time-dependent coefficients unique to each concentric layer, synonymous to the previously used A i and B i , but renamed to avoid overlap. The boundary conditions (4)-(7) once again apply, with the exception that z is replaced with r , and U 0 (t) occurs at some r = r 0 rather than r = 0. The intermediate steps are identical to those of (8)- (11), except with the instances of (z k−1 − z k ) replaced with ln r k−1 − ln r k . Equation (17) onward remain unchanged, as the differences in geometry are accounted for within the polynomial coefficients of (16). The radially decaying field magnitude in the i th layer is then given by

IV. GRADING CHARACTERISTICS OF MULTILAYERED COMPOSITES UNDER FAST-RISING IMPULSES
In this section, the results arising from the application of the model to FGMs are presented. The analytical nature of the model, coupled with the inclusion of arbitrary permittivity and conductivity in each layer, allows for rapid parametric and optimization studies to be performed. Thus, the model has the potential to be used in the design of layered or composite insulation systems under generalized time-dependent stresses, including for pulsed applications. Sections IV-A and IV-B follow to independently analyze the effects of permittivity and conductivity grading under fast-rising signals, respectively. Computational results gathered using a FE electrostatic solver package QuickField Professional version 6.4 [33] are also presented alongside the analytical model for validation. Quick-Field has the capability to conduct the transient electric field analysis, allowing direct comparison of (20) to numerically simulated results. It should be noted that due to the limitation of linearly spaced finite timesteps in QuickField, data points shown in Figs. 6 and 8 become irregularly spaced on a logarithmic scale. Simulation results are, therefore, not shown after 1 μs to maintain the visibility of other plotted data, but have been confirmed to continue following the analytical solution. It is further remarked that while experimental validation would be invaluable to ascertain these effects in practice, the fabrication of layered composites and the accurate measurement of intralayer fields under high speed impulses is nontrivial. An experimental investigation was, therefore, considered outside the scope of this theoretical study, though it would be of high interest to conduct such an investigation in the future.

A. Graded Permittivity
A 20-layer composite material was considered, with two different permittivity grading profiles under demonstration: 1) a linearly decreasing profile and 2) a "U"-shaped profile. Equations (24) and (25) have been developed to provide the parameter value at a given layer based upon the layer position, corresponding to the two profiles described above. Linear and quartic polynomials have been used to interpolate the parameter values for the two profiles, yielding  Here, p max and p min are the maximum and minimum desired parameter values (relative permittivity or electrical conductivity), and a = (1 − (−1) n )/2 is either 1 or 0 depending on the parity of n. If n is even, (25) holds only up to i = n/2, after which the resultant values for p must be reflected. Each layer in the composite is maintained at an equal thickness of h = 0.25 cm to give a total interelectrode gap of 5 cm. In the results presented in this section, the electrical conductivity of all layers was set to zero, such that the field relaxation in the bulk could be ignored, enabling the effects of permittivity grading to be isolated. The spatial variations of the relative permittivity for each of the aforementioned profiles are shown in Fig. 3, using ε max = 15 and ε min = 2. The choice of these values is based upon relative permittivity values for common insulating materials, for example, polymeric materials (cross-linked polyethylene (XLPE), polypropylene (PP), and ε r =2-3 [34]), ceramics (Macor, Alumina, and ε r =6-9 [35]), and an upper range (ε r = 12-15) possibly attained by some composite grading materials [36].
Using a particle-swarm-like optimization algorithm, the double-exponential wave-shaping parameters emulating a 50/500-ns impulse voltage (50-ns front time and 500-ns timeto-half value) were calculated, giving A 0 ≈ 1.1454, α ≈ 144.8861 × 10 4 s -1 , and β ≈ 49.8516 × 10 6 s -1 . This impulse voltage is applied to the HV electrode with a peak voltage magnitude of 150 kV, the full waveform of which is included in Fig. 4 (dashed line). The two additional waveforms in this figure are used later, in Section IV-C. The analytically predicted field distribution developed inside the composite is  shown with overlaid simulation results in Fig. 5, while the transient field responses in layers i = 1, 5, 10, and 15 are included in Fig. 6. For the purposes of demonstration, only results from the planar geometry are presented in this section. The field response follows the same trends in a cylindrically symmetric case, with the main exception being the radial decay of the field. Excellent agreement is found between the analytical predictions of (20) and the numerically simulated solution. For materials with zero conductivity, the electric field inside the composite layers behaves as expected. Since there is no influence from conduction currents, the rise and fall characteristics of the electric field in each layer mimic those of the applied signal. The magnitude of the electric field strength in each case, however, is strongly determined by the relative permittivity of the corresponding layer, whereby the degree to which the field is reduced is positively correlated with its respective value of ε r due solely to polarization. The field is, therefore, rejected from the high-permittivity layers toward the lower permittivity layers.

B. Graded Conductivity
To investigate the influence of electrical conductivity grading, a complementary study was conducted for a σ profile, but with identical ε r for all layers. In this case, the material  with the highest electrical conductivity was placed adjacent to the electrodes, in regions where the field typically requires reduction. The relative permittivity in all layers was set equal to ε r = 2, which reflects a typical value found for many polymeric insulators commonly used in engineering applications. The geometry and signal attributes remained the same as before, with the exception that now the electrical conductivity was graded following the profiles described by (24) and (25), with σ min = 10 −14 S/m (approximate range for common engineering polymers) and σ max = 10 −4 S/m (possible range attained by commercially available conductive grading tapes [37]). Once again, the analytical field distributions and transient responses using the linear profile are shown in Figs. 7 and 8, along with the numerical solution computed from QuickField.
With nonzero electrical conductivity, the field relaxation characteristics of the various layers were shown to influence and modify the transient response of the composite. Both the peak magnitude of the electric field and the peak time at which this maximum occurs in any layer showed strong dependence upon the electrical conductivity in all other layers. Analytically, this behavior follows from the n − 1 additional time constants arising from the characteristic polynomial and captures the complex relationship between the different relaxation times, which govern the dynamics of interfacial surface charge accumulation due to the Maxwell-Wagner effect. Under the linear grading profile, the direction of the field was also observed to reverse near the wave tail, at around t = 1 μs. This is believed to be the result of decaying surface charge at the interfaces between dielectric layers. The time required for the surface charge to dissipate at any interface was shown to be a complex function of the permittivity and conductivity distribution of the entire composite, which, in general, will not be equal for all layers. Therefore, on the falling edge of the impulse, the field in the (i + 1)th layer may momentarily be in the opposite direction to that of the i th layer due to the differing rates of surface charge decay at the i th layers' interfaces. The field then gradually decays back to zero, as the levels of surface charge at all interfaces decay to zero. Under the present linear grading profile, the field in each layer, therefore, reverses polarity consecutively, starting from i = 1 up to i = 20 in sequential order.

C. Effects of Wave Shape
Under combined permittivity and conductivity grading, the effects of the applied wave shape on the transient field response of the composite were also studied in this work. A wide variety of pulse shapes are utilized across various pulsed power applications [23], [24], [25], [26], [27], [28], which can range from picoseconds to microseconds in front time. As demonstrated in Section IV-B, the presence of layers with nonzero conductivity can substantially modify the transient behavior of the developed field inside the composite. Therefore, it is of great interest and high importance to characterize how a given composite would react under different impulses, particularly for applications, which may be sensitive to the wave shape, such as PEF treatment [25], [26]. In addition to the 50/500-ns signal previously characterized, we further introduce a 0.5/5-μs impulse and a standard 1.2/50μs lightning impulse according to IEC 60060 standards [38], as was shown in Fig. 4. These specifications were chosen to represent waveforms which are increasingly slower rising than the original (50/500 ns) waveform. The peak voltage remained at 150 kV, and the corresponding wave-shaping parameters are tabulated in Table I.
The combined grading scheme follows that of the "U"-shaped profile for both relative permittivity and conductivity, with the same values as used in Sections IV-A and IV-B. Fig. 9 plots the transient field responses for i = 1 and 10, corresponding to the positions where the field exhibits the most significant reduction and enhancement, based upon the choice of grading profile.
It should be noted that, due to the symmetry of the grading profile about the interface of the 10th and 11th layers, the field responses were found to be mirrored, such that i = 1, 20 or i = 2, 19 have identical responses. Evidently, for all three impulsive wave shapes, the fields at the electrodes are effectively reduced, each with a similar peak reduction Fig. 9. Transient field response for the layers i = 1 and 10 in the 20-layer composite under combined permittivity and conductivity grading, following the "U" profile and under energization by the three waveforms described in Fig. 4 and Table I. of ≈85%. The successful redistribution of the field to the middle of the composite material is also clear; however, the peak field magnitude which was developed differs significantly between the various applied waveforms. Around 10% higher peak field is estimated for the 0.5/5-μs signal compared with 50/500 ns, while a substantial difference of around 40% is exhibited for the 1.2/50-μs impulse. The likely causes of such wide variation are the magnitudes of the composite relaxation times relative to the applied impulse time constants. For the system studied here, system time constants arising from the characteristic polynomial range from a maximum of around 23 ms to a minimum of around 1.3 μs, with most of them existing between 1 and 10 μs. For fast input signals with short front times (e.g., 50/500 ns), the inability for the system to react to the rapidly changing external field results in a reduced maximum at the center of the composite, as there is insufficient time for the field to be excluded fully from the outer layers. This is consistent with Fig. 9, which shows a higher peak field in layer 1 for faster signals. To the contrary, when the input front time is similar to or longer than the above range (e.g., 1.2/50 μs), the dielectric layers are able to respond to a greater extent and redistribute the field inward on a similar timescale to that required for the applied field to peak. Therefore, a much higher degree of peak field enhancement at the center and stronger field reduction at the electrodes are observed. It follows, therefore, that this effect would likely have the greatest impact when the relaxation times of the composite are significantly longer than that of the characteristic timescales of the applied impulse.
It has, therefore, been shown that the grading effectiveness of a layered composite under impulsive stress strongly depends upon the transient characteristic of the applied signal and of the composite itself. Consequently, for the purposes of insulation design for pulsed power systems, the parameters of a layered composite system should be designed with careful consideration given to the applied impulse characteristics and material choice, if the desired performance is to be achieved.

V. APPLICATION TO GAS-INSULATED SYSTEM SPACERS A. Field at the Gas-Spacer Interface
As suggested in Section I, one possible solution to allow for the continued miniaturization of various components in HV gas-insulated equipment [e.g., gas-insulated transmission Fig. 10. Field distribution near disk-type GIL spacer for (a) uniform spacer material and (b) εand σ-graded material following "U" profile. Images were taken when the field enhancement at the interface was at its maximum. Dashed contour depicts the path of measurement for Fig. 11 (on the side of the gas).
lines (GILs)] relies on functionally graded solid dielectric spacers. Most importantly, the electric field distribution at the gas-spacer interface-particularly at the triple junction formed among spacer, gas, and electrode-essentially dictates the breakdown strength of the system, due to the high level of field enhancement that can occur within this region. Graded spacers have been shown to successfully reduce the electric field enhancement factor at triple junctions, which reduces the probability of electrical breakdown and subsequent system failure.
The model developed in this study does not explicitly consider the gas-spacer interface; however, this follow-on section shows that the field magnitude and the distribution at the gas-spacer interface can be roughly approximated to a reasonable degree by the present analytical model, thereby providing a valuable design tool for the analysis of gas-insulated components under transient stress. One may consider a radially symmetric disk-type spacer employed in GIL. This spacer can be approximated using the concentric cylindrical geometry of Section III-D and Fig. 2. This geometry is a simplification, particularly for some GILs used in power equipment, which consist of considerably more complex geometries. However, for spacers that are close to disk-like, e.g., as shown in [39] and [40], or for small-scale nonstandard systems used in some pulsed applications, which continue to employ simple disklike geometries, they can be considered well represented by the present model. Fig. 10 encloses the FE computed field distribution surrounding a simplified model of a disk-type spacer, with a radial dimension of r n − r 0 = 5 cm, an axial thickness of 3 cm, and with a conductor radius of r 0 = 2 cm. The 150-kV, 1.2/50-μs lightning impulse described in Table I was applied. Fig. 10(a) was generated with a homogeneous spacer material with identical properties in each layer of (ε r , σ ) = (2.3, 10 -14 S/m), contrasting that of Fig. 10(b), which was functionally graded following the 20-layer "U"-shaped profile as before, for both relative permittivity and electrical conductivity. E TJ denotes the maximum field magnitude at both the spacer-gas-electrode triple junctions. The grading effect is evident in Fig. 10, with the net effect of the electric field being redistributed toward the middle of the spacer where permittivity and conductivity are at their lowest. As such, the field enhancement at the triple junctions near the HV and the ground electrodes was reduced. The maximum recorded value of E TJ with an ungraded spacer was ≈14 kV/mm at the anode and ≈6 kV/mm at the cathode, compared with the respective values of ≈3 and ≈1.3 kV/mm when the spacer was graded, corresponding to ≈78% field reduction at both points. The field distribution at the gas-spacer interface (on the side of the gas) is shown in Fig. 11 (dashed line), which is compared with the analytically predicted field distribution modeled by (20) (solid line) in the spacer bulk. At any time, t, one may observe that the bulk field is a reasonable approximation of the field at the interface, particularly with regards to the shape of the redistributed field. While the analytical model does not provide exact field values at the interface, the general variation of the graded field is able to be approximated spatially and temporally. The model, therefore, provides a possible method for the rapid estimation of field redistribution at FGM-gas interfaces in GIS or other similar layered systems under time-varying stress, which may help to accelerate the design process of novel insulation technology. There are, of course, limitations to such an analytical approach. For example, the 2-D polar approximation would likely become inaccurate, as the thickness of the spacer decreases, since the derivation is limited by the assumption of invariance along the z-direction.
Despite this, the model has potential to be an important tool for the design of other pulsed power systems where the characteristics of the transient field are important, such as in PEF chambers. In these cases, the model may be used to estimate the transient field in an existing system, or be used to inform material choice to modulate the applied field with the aim to attain the optimal wave shape for a specific application.

VI. CONCLUSION
By employing an analytical approach to multilayered dielectric composites, this article has shown that the mathematical solutions exist for n-layer composites in 1-D planar and cylindrical geometries with arbitrary permittivity, conductivity, and layer thicknesses. Closed-form solutions have been shown to exist for up to five layers, while procedures for the solution when n > 5 have additionally been outlined. As such, the electric field can be computed without the need for FE software, allowing parametric and optimization studies to be conducted rapidly. Using the developed model, the analysis of a 20-layer composite was conducted under the application of fast-rising impulsive voltages. The transient response of the electric field inside the composite has been shown to be governed by the complex interplay between system relaxation time constants and input signal characteristics. The model has further been applied to FGMs for GIS spacer design, demonstrating the ability for the model to reasonably approximate the redistribution of the electric field at the gas-spacer interface of a graded disk-type spacer under time-varying energization.
Considering this, the model may be a valuable tool for the rapid estimation of field redistribution in layered dielectric systems, which may help to accelerate the design process of novel and nonstandard pulsed power systems when used alongside existing, more complex, modeling methods.
In general, much remains to be investigated regarding the influence of individual dielectric relaxation times on the overall response of the composite. The present model provides a convenient method to explore a vast range of composite configurations and to rapidly characterize their behavior, and due to its generality, its application is far from limited to FGMs or impulsive signals. The s-domain solutions may be modified for any arbitrary input waveform, and the composite thicknesses are also variable. With this flexibility in mind, the model may be of high interest to research in a number of other fields, including microbiological cell modeling, PEF treatment chamber design, multilayered capacitor design, insulation for more-electric aircraft, or for high energy density storage devices. However, there exist clear limitations to the present model, examples including the use of simple 1-D geometry, and the neglecting of σ (T, E) dependence, which may become important in high-field or long-term steady-state applications. In future work, it would be of great interest to investigate the effects of nonuniform conductivity in combination with layered dielectric material in order to further assess the suitability of FGM usage in environments at elevated temperatures or fields. Future experimental work would also be a significant step forward, both for model validation purposes and for advancing the techniques for the practical realization of multilayered composites.