Introduction. Electric field grading, or stress control, refers to the technique of reducing local enhancements of the electric field in various devices [1], [2]. As a part ...
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Nonlinear Resistive Electric Field Grading Part 1: Theory and Simulation Key words: electric stress control, nonlinear field grading material, finite element simulation, high voltage cable accessory Introduction Electric field grading, or stress control, refers to the technique of reducing local enhancements of the electric field in various devices [1], [2]. As a part of insulation coordination, field grading is crucial for making products competitive, particularly today when voltage levels are increasing, sizes of components are shrinking, installation logistics are an issue, and the demand for integration of different functions in a single device is steadily growing. Design criteria such as cost, safety issues and low electric fields and temperatures are often contradictive and must be balanced. For example, smaller insulation thickness leads to lower material costs and lower temperatures but to higher electric fields, which can lead to electric breakdown, particularly at critical regions such as interfaces or triple points. Appropriate field grading can help attain or improve and optimize a design that outbalances such criteria. Field grading methods fall into two main classes: i) capacitive field grading, e.g., geometrical electrode grading with appropriate shape of conductive parts, refractive grading with high-permittivity materials, and condenser grading with integration of metallic elements; and ii) resistive field grading, using special materials with appropriate current-field characteristics. This simple classification is based on whether the displacement, i.e., capacitive current, or resistive current dominates the field grading mechanism. Of course, at finite frequencies a combination of both capacitive and resistive field grading is active. Also in certain cable terminations where nonlinear resistive field grading tubes are used in combination with capacitive stress cones. The main task during product development of many electrical devices is to identify the best field grading technology for a given application. The result depends, among other things, on the maturity of the field grading technologies. In the following we focus on nonlinear resistive field grading, mainly because the maturity of this technology has considerably progressed during the past years for both ac and dc applications, enabling applications of increasingly higher system voltages. Based on conduction currents rather than displacement currents, resistive field grading can produce a considerable amount of heat and is thus apparently not suitable for use in HV devices.
November/December — Vol. 26, No. 6
Thomas Christen, Lise Donzel, and Felix Greuter ABB Switzerland, Corporate Research, BadenDättwil, Switzerland
Nonlinear resistive field grading permits the optimization of designs of ac and dc accessories in medium and high voltage systems. This first article of two introduces the theoretical basics required for a successful application of this modern technology.
This was a main reason why nonlinear resistive field grading technology was not used in HV ac applications and was restricted to medium voltage until recently [3]. However, the maximum electrical stresses often occur during short times only, e.g., during impulse type-tests, where Joule heating is irrelevant. Capacitive field grading for these cases leads to an oversized design for nominal operation conditions. It will thus be of benefit if nonlinear resistive field grading materials can be used, mainly because of a resulting smaller design. However, this requires that the field grading material acts only on these rare events of maximum short-time stresses and that at nominal stresses, it remains insulating with low losses. As will be discussed later, the main prerequisite for this is a strong field dependence, or nonlinearity, of its conductivity. Besides high field stresses that are strongly localized in time, another general problem in HV devices are field enhancements that are strongly localized in space, e.g., at electrode edges, corners, and protrusions. Again, appropriate nonlinear resistive
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field grading can be used to decrease this field enhancement, provided that the integral power production in the small critical region is uncritical for destructive thermal effects. Properly designed nonlinear resistive field grading materials are active only when necessary, e.g., during test conditions, overvoltages, or in spatially localized regions. This holds not only for ac applications but similarly for HV dc applications. Those are also tested with impulse, switching, and sometimes even ac voltages. At dc voltages, moreover, linear resistive field grading at nominal stresses can play an important role, as will be shown below. The goal of this first of two articles on nonlinear resistive field grading is to provide a basic understanding of the behavior of field grading materials with strongly field-dependent conductivities. A subsequent article will discuss the physical properties of these materials and their processing, and compare different types of field grading materials. Both issues are prerequisite knowledge for a successful application of nonlinear resistive field grading in the development of electrical devices such as ac and dc cable accessories, motors, or generators [1]–[6]. The next section recalls the basic behavior of the electric field distribution in solid media and shows that resistive field grading is related to space charge formation. Afterward, the required electrical properties and the behavior of nonlinear resistive field grading materials will be discussed, including a combined description of nonlinearity and frequency dependence, as well as Joule heat production. Finally, simulation results are presented to illustrate application to real devices under realistic electric stress conditions.
The Behavior of the Electric Field We consider cases in which the electric and magnetic fields E and H are governed by the following two Maxwell equations in media,
¶ (eE ) Ñ ´H = j + ¶t Ñ × (eE ) = r,
(1 - 2a )(1 - r / r ) . f =
(1)
where inductive effects can be neglected (Ñ´ E = 0). Here, ε = ε0εr is the dielectric permittivity, j is the electric conduction current density, and ρ is the space charge density. The first equation states that the total current density,Ñ ´H, is the sum of conduction and displacement currents. The two limit cases of capacitive and resistive field distributions are associated with Ñ × (eE ) = 0 and Ñ × j = 0, respectively. The former is related to the absence of space charge, and the electric field E = -Ñf is obtained from solving the Laplace equation Ñ × (eÑf) = 0 for the electrical potential f. Resistive field grading, on the other hand, is associated with a change of the capacitive field distribution and thus involves space charge. Indeed, Ñ × ( j + ¶ (eE ) / ¶t ) = 0
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and r = Ñ × (eE ) = 0 cannot be satisfied, in general, at the same time. The relations between material quantities and field can be obtained from theoretical solid state physics for bulk materials, interfaces, or contacts. Without going into details, we mention that a full theory may consist of additional dynamic variables (electron, hole, and ion densities) and involve a number of additional time scales (generation-recombination, carrier transit, and trapping) and length scales (e.g., Debye length). In this article, however, we assume for simplicity that the materials are isotropic and can be described by constant permittivities and a local, instantaneous, and single-valued functional relation between current density and electric field, j = s(E )E , where σ(E) is the field-dependent conductivity with E = E . One must be aware that these assumptions may oversimplify electric conduction, and it might be necessary to incorporate intrinsic temporal and spatial degrees of freedom of the conduction mechanism, e.g., charge carrier densities. A simple way to proceed in such cases is presented in equation 3 of reference [7]. Let us first illustrate nonlinear resistive field grading by one of the most simple examples: two spherically concentric electrodes with inner and outer radii, r1 and r2, respectively, filled with a linear or nonlinear material obeying j µ E a-1E , and stressed by a dc voltage V0. Here, α describes the nonlinearity (α = 1 refers to a linear material), and the ratio r1/r2 characterizes the geometrical inhomogeneity. The case r1/r2 1) and the Laplace field value (with α = 1) at r1, turns out to be -1
1
1-2/a
(r2 / r1 )
-1
2
(2)
Note that f is a function of the nonlinearity α and the geometry, i.e., r1/r2. Illustrative examples are plotted in Figure 1 for r1/r2 = 4·10−2 and 4·10−5. Almost full field grading is reached at α >10. At α = 5, field grading is already sufficient for practical applications, for r1/r2 not too small. For fixed radii and α→∞, it holds f→r1/r2, which reflects that the field is equal to the average field, E(r) ≡ V0/(r2 − r1). This is a simple case in which field grading is limited by the geometry, which addresses an important aspect of global field grading to be discussed below. In the case of r1/r2→0, i.e., at a localized and huge field enhancement in an otherwise large geometry, one has f→0 for α→∞. Such huge field enhancements often have to be reduced practically by many orders of magnitude, which is only feasible for sufficiently large α. What sufficient means in this case is the main question regarding what will be referred to below as local field grading. For the example in Figure 1, one observes that for IEEE Electrical Insulation Magazine
Figure 1. (a) Normalized field distribution for spherically concentric electrodes (inset) with r1 /r2 = 0.04 and different nonlinearities (red: without grading, blue: moderate α, black: ideal grading with very high nonlinearity). (b) Field reduction factor f for two different r1 /r2 ratios as a function of α. The encircled regions indicate α-values above which an increase will not lead to significant further improvement. α >15 to 20, no significant further improvement of field grading can be obtained. This finding is particularly relevant for field grading material development, because the higher the nonlinearity, the more difficult it is to develop and produce a corresponding field grading material. Nevertheless, as will be discussed in the following, there are further effects and requirements that affect the optimum value of α. The steady-state space charge can be calculated directly from the electric field E(r) by using the Poisson equation. For instance, in the α→∞ limit where the field becomes constant, the space charge between the spheres is distributed according to ρ = 2εE/r. We need also to understand the dynamic behavior of the electric field. For this, it is instructive to combine the two equations in (1), which gives the charge continuity equation in the form
¶r r = -Ñ × j = + j ×Ñ ln(tM ), ¶t tM
(3)
with τM = ε/σ. In linear materials, where σ is field independent, τM characterizes the relaxation time for exponential convergence of space charge toward its equilibrium value. In addition, 1/τM can be interpreted as the characteristic frequency for crossover between resistive and capacitive field distributions at time-harmonic voltages. For general nonlinear materials, τM characterizes an order of magnitude of the characteristic time scale; the exact relaxation time associated with exponential decay of small deviations from the steady state is related to the differential conductivity, dj/dE. In practice, τM can be understood as a characteristic time scale for the decay of transient fields after immediate voltage switching. In summary, capacitive field distributions are present when the displacement current is dominant and space charge has insufficient time to form, i.e., for times t > τM and r » eE ×Ñ ln(tM ). The latter relation indicates that stationary space charge can be associated with a spatial τMgradient in the direction of the electric field, which can be of relevance for functionally graded materials with specifically designed spatial conductivity gradient.
Required Material Properties The resistive field grading mechanism works by bringing a critical high field region to a conductive state such that space charge forms there and creates a counter field, which reduces the field enhancement. A nonlinear resistive field grading material has thus a field-dependent conductivity σ(E) that increases strongly from a low conductivity value, σ(0), to a high value in a narrow field region ΔE around a value Eb. The value of the switching field Eb indicates where field grading becomes active. The order of magnitude of σ affects, as discussed, the time scale for the activation, and ΔE/Eb is related to the nonlinearity of the field grading material. It is important that the change from the highly resistive to the highly conductive state occurs reversibly. This ensures repeatability of activation and low losses. Hysteretic switching must not occur (e.g., as for an S-shaped current-voltage characteristic), which may lead to, beside large losses, unwanted effects such as current-filament formation due to multistability. The field value in the critical region eventually adjusts itself to a value near Eb via space charge build-up. The smaller ΔE/Eb, i.e., the higher the nonlinearity, the closer it is to Eb. For given specification requirements of an electrical device, e.g., defined by the applied test voltage, and the maximum acceptable electric field values at specific locations and under specific conditions, one can estimate the region in the E-σ plane
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Figure 2. Qualitative illustration sketch of how regions are determined in the E-σ plane for the σ(E) curve of an ideal nonlinear resistive field grading material: (a) ac applications, (b) dc applications.
where σ(E) should lie for an effective field grading material (Figure 2). This issue is crucial if one wants to design the electrical properties of a field grading material; the material science aspect of how this is done in practice will be described in the second article. Consider first ac applications with angular frequency ω of the time-harmonic voltage. At moderate field stresses EN associated with average field values at nominal voltages, space charge formation should be slow compared with the ac frequency. However, it should be fast at fields of the order of the ac design field EAC (>>EN). This ensures low Joule heating at nominal fields but grading of critical field enhancements at higher fields. It implies ωτM(EN) >> 1 (i.e., capacitive regime, small loss angle) and ωτM(EAC) > means roughly a factor of about 100, to be safe (the loss angle of a resistive field grading material at nominal fields should be definitely smaller than 10%, preferably smaller than 1%, which might be difficult to reach). At impulse voltage, the design field EP is usually larger than EAC because of the short duration τP of the impulse (τP is typically milliseconds for switching surge or microseconds for lightning impulse). The field grading condition for impulse is τM(EP) > εω ≈ 3⋅10−8 S/m. One might thus decide to take σ(EAC) ≈ 10−6 to 10−7 S/m. This would imply an estimate for the nonlinearity of α > 1 + ln[σ(EAC)/σ(EN)]/ln[EAC/EN] ≈ 13 to 17 in the interval EN < E < EAC. The material will then have a switching field Eb of about 1.5 kV/mm. By further proceeding in this way for surges and impulses, with their own time scales and design fields, one can figure out the regions where the conductivity should lie, as is sketched in Figure 2. As mentioned, for dc voltage, the low field requirement for the maximum conductivity is often related to the leakage current of the component, whereas the minimum value depends on the resistivities of the adjacent insulation materials. This will be discussed in more detail below in the context of a dc cable joint.
The Field Grading Behavior Let us define a prototype nonlinear current-field relation and study how it will execute field grading in principle. The four main electrical parameters are the relative permittivity εr, the small-field conductivity σ(0), the switching field Eb, and the nonlinearity α. In the literature there are various fit functions for σ(E) to experimental data. An example is σ = σ0(1 + (E/Eb) α−1 ) with a characteristic conductivity scale parameter σ0, which is here equal to σ(0). Because we are mainly interested in the effect of the nonlinearity α, we will assume a simple power law,
æ E ö÷a-1 j = s0 ççç ÷÷ E , çè Eb ÷÷ø
(5)
and neglect the small constant σ(0), except for linear materials with α = 1. The limit α→∞ describes an ideal insulator–conductor transition. Because the final aim of field grading is to prevent failures (flashovers, punctures, thermal runaway) or degradation (partial discharges, tree formation) by controlling the electric field strength at critical locations, it is helpful to distinguish for practical purposes between the following: • Local field grading: Suppression of huge field enhancements (by orders of magnitude) in localized spatial regions, e.g., at very small structures, close or even at the surface of the field grading material, e.g., at semicon edges, corners of slots at machine end windings, and electrode protrusions. Local nonlinear resistive field grading may act here for long time periods, e.g., at nominal voltage, provided the integral Joule heating is small due to the limited volume of the active region. • Global field grading: Equalization of the electric stress, typically down to 10 to 20%, in large spatial regions, sometimes even far away from the field grading material, e.g., at an insulator surface of a cable termination or a bushing. For ac devices, global nonlinear resistive field grading is restricted to short time periods because of the typically large losses of active nonlinear resistive field grading materials, which can lead to harmful heat production. Fast reaction time at impulse stresses requires a large conductivity. For dc applications, global resistive field grading is very important also for long
November/December — Vol. 26, No. 6
time periods, i.e., at nominal dc voltage and not only for surge or pulse tests, and usually acts in the linear (or low-α) low-leakage conduction regime of the field grading material. Linear resistive dc field grading acts comparatively slowly, such that the field grading can be poor during the initial phase of, for example, switching-on a dc voltage. Distinction between local and global field grading is only a rough classification. Local and global aspects are often entangled in practical cases. Both cases will now be discussed in more detail.
Local Resistive Field Grading Consider a strongly localized large field enhancement, where the nonlinear resistive field grading material is conductive and displacement currents can be neglected. It is instructive to analyze singular geometries, i.e., points, lines, edges, or corners, with zero radius of curvature. The electric field, E(r), diverges as the distance r from the singularity goes to zero, r→0. Of course, real edges and tips always have a finite radius of curvature R, and the limit r→0 is not physical. However, the maximum field in a real configuration can then be estimated from the results by taking the value E(r = R). For a point electrode, sphere with radius →0 in space dimension d = 3, and a line electrode, cylinder with radius →0 in d = 2, in a linear, uniform medium (α = 1), the electric field is Laplacian, i.e., the electric potential satisfies the Laplace equa tion, and diverges as E ∝ 1/rk with k = d − 1. FromÑ × j = 0, one obtains, for a nonlinear resistive field grading material with current density according to (5), k = (d − 1)/α. This already quantitatively indicates the beneficial decrease of the divergence as the nonlinearity α increases. We will now discuss a few further practically important cases such as edges and corners. The field divergence depends then on the angle θ as sketched in Figure 3. For edges with opening angle θ (Figure 3a), the Laplacian field diverges as k = 1 − 1/ (2 − θ/π) [8]. For instance, θ→0 may describe a thin semicon edge of a cable end, and θ→π/2 can be associated with a motor ground wall end. The respective Laplacian fields diverge as 1/ r1/2 and 1/r1/3, respectively. For cones (Figure 3b) with opening angle θ, one has k = 1 − 0.5/ln(4/θ) [8]. For instance, the end of a thin needle can be described by θ→0, or a general electrode protrusion can be described by an angle 0 < θ < π. The end of a thin needle is the worst case with 1/r-field divergence. To see how nonlinear resistive field grading materials with nonlinearity α can reduce these field enhancements, we have calculated the fields numerically by solving Ñ × j = 0. The electric field as a function of the distance x from a needle tip is shown in Figure 4a for various values of α. For the needle and an edge, the field value at a fixed distance is plotted as a function of α in Figure 4b. From the results, one can conclude that, as a rule of thumb, α ≈ 10 to 20 is appropriate for local grading of huge field enhancements. For a material with small α < 10, a slight further increase of α may still lead to a large effect, whereas the benefit of a much higher nonlinearity than α ≈ 20 is marginal. 51
Figure 3. Edge (a) and cone (b) shaped electrodes with angles θ.
Global Resistive Field Grading Global field grading refers to cases in which not only the vicinity of a small geometrical inhomogeneity, such as an edge or a corner, but the total geometry or even the topology has to be considered. One example was discussed in Figure 1, namely when the outer radius r2 starts to play a role. Another example is associated with a decoupling of different insulation systems in dc accessories, as will be explained later in Figure 10. For time-dependent voltages such as ac or impulse, global field grading generally also involves the displacement current, ¶ (eE ) / ¶t. Whereas for high local field enhancements discussed in the previous section, this term could be neglected, a discussion of global field grading has to take into account the total cur rent density, sE + ¶ (eE ) / ¶t. The whole spatial domain consists then of different regions, those where resistive field grading is active and resistive field distributions dominate, and those
where it is inactive and capacitive field distributions dominate. As a first approximation, the former ones are the regions where the Laplacian field values, EL (x ), are larger in amount than the switching field, Eb. Due to the decrease of the field strength by the action of the field grader in this region, the field in its vicinity will increase and lead to a self-consistent extension of the region where the field grader is active. The size of the eventually formed active region is characterized by a penetration depth, Δ, which will play a crucial role in the following. An important application example of global field grading is in cable accessories such as terminations or joints. A typical topology is shown in Figure 5. A detailed discussion is given in [9]; in the sequel, the most important findings reported in this reference are summarized. For an arbitrary nonlinearity α and frequency ω, the penetration depth Δ is obtained from the implicit equation
Figure 4. (a) Electric field versus axial distance x (solid curves) from the needle tip at 1 V and for the configuration shown in the inset of (b). Dashed line indicates the slope −1 of the Laplace field near a tip. (b) Electric field versus α at 0.5 mm from the singularity. Red: tip of a long needle as in (a) (inset with cylindrical symmetry); blue: edge of thin film (inset with 2d Cartesian coordinates). 52
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1) and capacitive (ωτFG >> 1) cases can be associated with penetration depths D = DV / w and D = DV tFG =: De , respectively. The capacitive value, Δε, is generally a lower bound for the penetration depth. Second, in the limit of purely resistive field grading (ωτFG
