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Oct 8, 2012 - Simulation in Insulating Materials. Daomin Min and Shengtao Li. State Key Laboratory of Electrical Insulation and Power Equipment,.
IEEE Transactions on Dielectrics and Electrical Insulation

Vol. 20, No. 3; June 2013

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A Comparison of Numerical Methods for Charge Transport Simulation in Insulating Materials Daomin Min and Shengtao Li State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi'an, Shaanxi, 710049, China

ABSTRACT Bipolar charge transport (BCT) model has been widely used to simulate time/space evolution of space charges in insulating materials. The BCT simulations are performed to investigate the relationships between space charge accumulation and conduction, electroluminescence (EL), charge packet formation, electrical breakdown, and surface potential decay (SPD) properties. Accordingly, the charge advection-reaction equation that contains shocks or high gradient regions should be solved by highly accurate and stable numerical methods to obtain high resolution. We use Runge-Kutta discontinuous Galerkin (RKDG) method and finite differential weighted essentially non-oscillatory (WENO) method to resolve the charge advection-reaction equation. Then, we calculate the SPD properties and space charge profiles of corona charged low-density polyethylene (LDPE) at various initial surface potentials. The simulated results of the two schemes are compared with analytical SPD results, and also compared with each other. It is found that the simulated SPD curves of RKDG and WENO in the case of single carrier injection are both consistent with the analytical results. Moreover, in the case of both single carrier injection and bipolar carrier injection, WENO scheme is more accurate than RKDG scheme at a given spatial discretization. Index Terms — Bipolar charge transport, LDPE, RKDG, surface potential decay, weighted essentially non-oscillatory (WENO).

1 INTRODUCTION1 THE charge transport, electroluminescence (EL), electrical breakdown, and surface potential decay (SPD) properties of insulating materials can be affected by space charge formation and migration processes. Current-voltage characteristics will change from Ohmic regime to space charge limited current (SCLC) regime, when the applied voltage exceeds a critic value [1, 2]. Measurements by using pulsed electroacoustic (PEA) technique demonstrated that no space charge was formed in polymeric materials at low electrical stresses, however, charges began to be accumulated in the materials when the applied electric fields were higher than a threshold value (FT) , e.g. FT ~107 V·m-1 for low density polyethylene (LDPE) at 20 oC [3, 4]. Moreover, time dependent current properties are also influenced by space charge dynamics. For theoretical analysis, transient space charge limited current may be considered at high applied voltages [5]. For numerical simulation, we can adopt bipolar charge transport (BCT) models based on different physical assumptions, such as shallow trap controlled carrier mobility with charge trapping/detrapping in deep traps [6, 7], charge hopping in a single hopping level [8], or charge hopping in an exponential distribution in energy of traps [7]. In addition, EL in insulating materials submitted to DC [9] and AC [10] electrical stresses, especially the recomManuscript received on 8 October 2012, in final form 16 April 2013.

bination controlled EL, has been successfully investigated by EL spectrums and BCT simulations. It was found that electron-hole recombination could well interpret the observed EL results of polyethylene [9, 10]. The formation and migration of charge packets can result in the electrical breakdown of insulators. Matsui et al measured the time evolution of space charges in LDPE under DC stresses, and found that charge packets would occur when the applied electric fields exceeded about 108 V·m-1 [11]. The charge packets can distort the electric field distribution in the samples. When the Laplace electric fields were more than 2×108 V·m-1 or the maximum electric fields were more than 5.2×108 V·m-1, breakdowns were observed in the PEA results [11]. Matsui et al divided the zone behind and ahead of the charge packet into zone 1 and zone 2, and assumed that the conductivity in zone 1 was 14.4 times higher than that in zone 2 [12]. Then, they simulated the space charge formation and breakdown properties of LDPE, and found that the simulation results were consistent with the experimental results. Recently, the electrical breakdown behaviors of LDPE caused by charge packets were numerically studied by BCT models considered negative differential mobility [13, 14]. Furthermore, space charge and surface potential experimental results of insulators after charge deposition by corona or electron beam indicated that the SPDs calculated from space charge depth distributions were in good agreement with

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D. Min and S. Li: A Comparison of Numerical Methods for Charge Transport Simulation in Insulating Materials

those obtained by non-contact potential probe [15, 16]. Chen proposed to use the BCT model to analyze the SPD observations [17]. It was found that the bipolar charge injection and conduction could well explain the SPD crossover phenomenon [17, 18]. The BCT model has been widely used to investigate the space charge dynamics in insulating materials and its relationships with conduction, dissipation, breakdown, and SPD etc. It is significantly important to adopt highly accurate and stable numerical methods in the BCT simulations. Le Roy et al showed that several methods are not suitable to be used in the BCT simulations [19]. For example, FTCS and LAX-WENDROFF schemes are not stable, and first order upwinding scheme is strongly diffusive in the case of existing discontinuity or shocks. Accordingly, they proposed to used QUICKEST scheme with ULTIMATE flux limiter [19]. However, Tian et al pointed out that Runge-Kutta discontinuous Galerkin (RKDG) method is better than QUICKEST+ULTIMATE method to capture shocks in the space charge profiles [20]. Additionally, a finite differential weighted essentially non-oscillatory (WENO) method developed by Jiang et al [21] was utilized to resolve the charge advection equation, which showed high resolution [18]. Consequently, it would be very interesting to compare the accuracy of RKDG scheme [22] with that of WENO scheme.

2 BIPOLAR CHARGE TRANSPORT MODEL We consider a system consisting of an insulating material with the thickness of L and a grounded electrode, as shown in figure 1. A one-dimensional coordinate, x, is set up for the sample. x=0 corresponds to the charged surface of the insulator, and x=L corresponds to the interface between the material and the grounded electrode. Charges are assumed to be just deposited on the surface of the corona charged material. We assume that the electric field at xtT, dS(t)/dt decreases in time, so the surface potential decays slower and slower. The SPD curves at various initial surface potentials approach to each other after decaying for some time, however, no SPD crossover occurs in the UCT simulation.

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the calculated and analytical surface potential, respectively, and Scal can be obtained by RKDG or WENO. SPCRKDG and SPCWENO are the space charge density in the sample during SPD process, and they can be mobile charge density, trapped charge density, or net charge density. M and N are the number constants that are chosen for error evaluation.

Figure 3. Comparisons of analytical SPDs and simulation results by WENO scheme of corona charged LDPE with various initial surface potentials in the case of single carrier injection without trapping.

Figure 4 shows the relative errors of analytical and simulation SPDs at various initial surface potentials. The relative errors decrease with increasing the spatial discretization element, N, especially for RKDG scheme, when N changes from 100 to 300. At N=100, the relative errors are larger than 4.7%, however, the relative errors are smaller than 1.3% at N=300. Additionally, at a given N, all the relative errors of SPDs by WENO method are smaller than those by using RKDG method. It means that WENO scheme is more accurate than RKDG scheme.

Figure 2. Comparisons of analytical SPDs and simulation results by RKDG scheme of corona charged LDPE with various initial surface potentials in the case of single carrier injection without trapping.

In order to check the accuracy of RKDG and WENO schemes, we defined the relative errors of calculated and analytical SPD results and the relative errors of space charge density profiles as the following. ErrorSP  ErrorSPC 

1 N

N

1 M

  SPC i 1

M

 

(ti )  Sana (ti ) 

2

( xi )  SPC WENO ( xi ) 

2

i 1

RKDG

cal S

(18)

(19)

where, ErrorSP and ErrorSPC are the relative error between calculated and analytical SPD results, and the relative error between space charge profiles simulated by RKDG scheme and that by WENO scheme, respectively. Scal and Sana are

Figure 4. Relative errors of analytical and simulation SPDs of corona charged LDPE with various initial surface potentials.

Since the surface potentials of the sample are the double integration of space charges, the relative errors of SPDs of RKDG and WENO schemes are caused by space charge resolutions. Figure 5 demonstrates the space charge pro-

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D. Min and S. Li: A Comparison of Numerical Methods for Charge Transport Simulation in Insulating Materials

files calculated by RKDG and WENO schemes at various decay times at the initial surface potential of -5 kV, when the sample is divided into 100 parts. It is clear that WENO method can capture the steep leading front of charge carriers better than RKDG. The positions of leading front of charges at t=50 s are 2.85×10-6 m and 2.80×10-6 m for RKDG and WENO, respectively. Nevertheless, the positions calculated by equation (B2) is just 2.5×10-6 m. When N is set as 500, we obtain very steep leading fronts, as shown in figure 6. This can explain why the relative errors of SPDs of the two schemes decrease with increasing N. If the discretization is not adapted (not enough cells), the simulated results are far from the analytical ones whatever numerical scheme. If there is a difference between simulated results calculated with a different number of grid points, it means that the number of cells is not high enough for the problem under study.

cur during SPD process [15, 46]. It is interesting to use the BCT model [17] to explore the SPD properties of corona charged LDPE. Consequently, we carried out SPD simulations by RKDG and WENO methods in the case of bipolar charge injection. The following equation is defined as the relative error between SPD results simulated by RKDG scheme and those by WENO scheme in the case of bipolar charge injection. ErrorSPBCT 

1 M

M

  i 1

RKDG S

(ti )  SWENO (ti ) 

2

(20)

where, ErrorSPBCT is the relative error of SPDs. SRKDG and

SWENO are the calculated surface potential obtained by RKDG or WENO schemes, respectively. Figure 7 shows the comparisons of SPD curves calculated by RKDG and WENO schemes in the case of bipolar charge injection, where N=300. The simulation results of RKDG and WENO at various initial surface potentials are in good agreement with each other. At the initial surface potentials from -1 kV to -8 kV, the relative errors of SPDs evaluated by equation (20) are smaller than 7.7% and 4.9% at N=300 and N=500, respectively, as demonstrated in figure 8. Since the magnitude of net charge density increases with increasing the initial surface potential, the shock capture ability of numerical method will affect its accurate. Consequently, the relative errors at high initial surface potentials are larger than those at low initial surface potentials.

Figure 5. Comparisons of net charge density profiles calculated by RKDG and WENO at various decay times. The initial surface potential is -5 kV.

Figure 7. Comparisons of SPDs of corona charged LDPE with various initial surface potentials, calculated by RKDG and WENO schemes, respectively. Figure 6. Comparisons of net charge density profiles calculated by RKDG and WENO at various decay times. The initial surface potential is -5 kV.

4.2 COMPARISON OF WENO AND RKDG METHODS IN BCT MODEL Experimental results of PEA indicated that bipolar charge injection from both the charged surface and the grounded electrode into the bulk of the material would oc-

SPD crossover phenomenon occurs in the BCT simulation results. The surface potentials of the charged sample with low initial potentials decay very slowly. When S0 is -1 kV, the surface potential just decays 287.5 V. However, the surface potentials with high initial values decay very fast. When S0 is -8 kV, the surface potential decays 7004.7 V. In figure 7, the threshold initial surface potential of SPD crossover is about -5 kV.

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In figure 9, the net charge density profiles calculated by RKDG and WENO methods are compared at various decay times. The initial surface potential is –5 kV, and the sample is divided into 300 parts. As can be seen in figure 9, the leading front of positive and negative charges calculated by WENO is steeper than that obtained by RKDG. After the electrons and holes encountering each other in the center of the sample, t൒50 s at S0=–5 kV, the differences between the net charge density profiles calculated by RKDG and those by WENO become smaller and smaller.

Figure 10. Relative errors of net charge density profiles calculated by RKDG and WENO schemes as a function of decay time, respectively.

Figure 8. Relative errors of analytical and simulation SPDs of corona charged LDPE with various initial surface potentials.

Figure 11. Relative errors of net charge density profiles calculated by RKDG and WENO schemes as a function of decay time, respectively.

5 CONCLUSIONS

Figure 9. Comparisons of net charge density profiles of corona charged LDPE with the initial surface potential of -5 kV, calculated by RKDG and WENO schemes, respectively.

We calculated the relative errors of net charge density profiles as a function of time by equation (19). Figure 10 and 11 demonstrate the relative errors between RKDG and WENO at N=300 and N=500, respectively. The relative errors at various initial surface potentials all decrease with increasing decay time. When the decay time is larger than the transit time of carriers, the variations between RKDG and WENO are very small.

Space charge dynamics and surface potential decay (SPD) of low-density polyethylene (LDPE) after charge deposition by corona discharging were simulated by bipolar charge transport (BCT) model. In the BCT calculations, we used Runge-Kutta discontinuous Galerkin (RKDG) and finite differential weighted essentially non-oscillatory (WENO) methods to resolve the charge advection-reaction equation. Boundary element method (BEM) was adopted to solve Poisson’s equation. In the case of single carrier injection, the SPD outputs of RKDG and WENO schemes were compared with analytical results, and the two methods both showed nice performances. Nevertheless, a too sparse discretization grid will lead to false results. Then, we compared the space charge profiles of RKDG and those of WENO. It was found that WENO could capture shocks or high gradient regions better than RKDG.

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When taking bipolar charge injection into account, we calculated the SPDs and space charge profiles by RKDG and WENO shcemes, respectively. The outputs of the two schemes were compared with each other, and the relative errors of SPDs and space charge profiles were smaller than 0.049 and 0.26 at N=500, respectively. The simulation results indicated that both RKDG and WENO are highly accurate when the grid mesh is dense enough. However, if the sample is very thick and the grid mesh is relative sparse, WENO scheme would be a better choice than RKDG.

APPENDIX A. ON THE ATTEMPT-TO-ESCAPE FREQUENCY The experimental results show that the attempt-to-escape frequency changes in a wide range (typically, 1010 s-1-1014 s-1). Watson estimated that the attempt-to-escape frequency of solvent-purified PS2 polystyrene was 3×1012 s-1 from the temperature dependence of charge decay results [47]. From the PEA results, Zhou et al found that the attempt-to-escape frequencies of additive-free LDPE were about 1.8×1013 s-1 and 3.5×1013 s-1 at the trap energies of 0.88 eV and 1.01 eV, respectively [48]. Moreover, the thermally stimulated current (TSC) results indicated that the attempt-to-escape frequency of polyimide was about 8.5×1011 s-1 [49]. It was also found that surface modifications and heat treatments on insulating materials could change the attempt-to-escape frequency [50, 51]. It means that the existing equations of the attempt-toescape frequency cannot satisfy the experimental results [7, 52]. Much more researches need to be done on the attempt-toescape frequency in the future. However, for a comparison study on numerical methods, it is acceptable to use the equation proposed by Boufayed et al [7]. B. ANALYTICAL SURFACE POTENTIAL DECAY WITH SINGLE CARRIER INJECTION From equation (1), we can obtain the time dependent surface charge density [29].

QS  t   QS 0  exp    t 

(B1)

where, β=  ATE exp   EST kBT  . β determines the decay rate of surface charges. In the case of single charge injection, only electrons on the bare surface are migrated into the bulk of the material and no holes are injected from the grounded electrode into the bulk. According to Poisson’s equation, electric fields between the leading front of injected electrons, xlf, and x=L will not be changed before the leading front reaching the grounded electrode. In addition, if we ignore the electric field effect on carrier mobility, the electron mobility will be a constant without charge trapping/detrapping processes. Consequently, the migration velocity of the leading front will also be a constant. Therefore, the transit time, tT, of injected charges in the material can be calculated by the Newton’s law, tT 

L2

0S 0

where, μ0 is the trap free carrier mobility in m2·V-1·s-1.

(B2)

Then, we can obtain the surface potential as a function of time at two separated regions. The equation (31) in [29] is amended. Region I: 0≤t≤tT

S  t   S 0 

2

2

 0  S 0   t   1  exp  2 t    0  S 0  (B3)   4  L  2  L 

Region II: t≥tT

S  t  

0 4

2  2  LF  L, t   S 0  (B4)  F L , t exp  2  t        2  L   

where, the electric field at x=L can be calculated by the following equation. F  L, t  

 x  exp    t   0 F  L, t   L 

S 0

(B5)

Therefore, we can calculate the surface potentials of charged insulators by equations (B3)-(B5) in the case of unipolar charge injection.

ACKNOWLEDGMENT D. Min thanks the China Scholarship Council (CSC) for providing the scholarship for visiting Ph.D. studentship supervised by Prof. Mengu Cho at Kyushu Institute of Technology (File No. 2010628049). This work is supported by the National Basic Research Program (973 Program) of China (Grant No. 2011CB209404) and Project by State Key Laboratory of Electrical Insulating and Power Equipment (Grant No. EIPE09107).

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D. Min and S. Li: A Comparison of Numerical Methods for Charge Transport Simulation in Insulating Materials Daomin Min was born in HuBei, China, in December 1985. He received the B.S. degree in electrical engineering in Chang'an University in 2007. Currently, he is a graduate student of Electrical Engineering in Xi'an Jiaotong University. He was a Ph.D. visiting student in the Laboratory of Spacecraft Environment Interaction Engineering (LaSEINE), Kyushu Institute of Technology (KIT) from 2010 to 2011 under the support of China Scholarship Council (CSC). His main research is charge transport properties of insulating materials.

Shengtao Li, (M'96-SM’2011), was born in Sichuan, China, in February 1963. He received the B.S., M.S. and Ph.D. degrees in electrical engineering, from Xi'an Jiaotong University (XJTU) in 1983, 1986, and 1990, respectively. He worked at Waseda University, Tokyo, Japan, as JSPS research fellow for 3 months in 1996, and did research at the University of Southampton, Southampton, UK, as a senior visiting scholar for 6 months in 2001.He was a Lecturer, Associate Professor, and Professor with Xi'an Jiaotong University, China, in 1990, 1993, and 1998, respectively. From 1993 to 2003, Prof. Li was a deputy director of the State Key Laboratory of Electrical Insulating and Power Equipment (SKLEIPE) in Xi'an Jiaotong University. Since 2003, he has been an executive deputy director of SKLEIPE. He received financial support from the National Science Foundation for Distinguished Young Scholars of China in 2006. He is an Associate Editor of the IEEE Transactions on Dielectrics and Electrical Insulating. His research interests include dielectrics and their application, insulating materials and electrical insulating.