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Improved space charge transport model in bi-layer dielectrics—considering carrier dynamic equilibrium

: The Maxwell–Wagner model and bipolar charge transport model both aim at describing the charge transport behaviours in dielectrics. The Maxwell–Wagner model performs well in calculating the polarisation current and electric field distribution, but it is too macroscopic to describe the microscopic behaviours of space charges. The bipolar charge transport model can well simulate the space charge accumulation, but it cannot quantitatively relate the microscopic transport behaviours with the macroscopic conductivities of dielectrics. Considering the shortages of the above two models, an improved charge transport model, named carrier dynamic equilibrium model, was proposed in this study to simulate the polarisation process of the low-density polyethylene/ethylene propylene diene monomer bi-layer dielectric by introducing a source term of carrier dynamic equilibrium. Effects of carrier mobility and non-equilibrium carrier lifetime on the simulation results were explored, and a comparison among the results of measurement and different models was also made. Compared with the Maxwell–Wagner model and bipolar charge transport model, the improved model has the best coincidence with measurements, which can provide an accurate reference for the design of high-voltage direct current insulation systems.


Introduction
In high-voltage direct current (HVDC) transmission systems, space charge accumulation has a significant impact on the distribution of electric field in dielectrics [1][2][3][4][5][6][7]. The electric field distortion caused by unreasonable space charges may trigger an electrical tree and even insulation breakdown. Numerical simulation is a good method to figure out the distribution pattern as well as the accumulation mechanism of space charges. There are two popular models, i.e. the Maxwell-Wagner model and the bipolar charge transport model, in simulating the space charge and electric field distributions in dielectrics. The Maxwell-Wagner model has been widely used to describe the polarisation behaviours in composite dielectrics since it was proposed [8]. The Maxwell-Wagner model is based on the macroscopic electrical parameters, i.e. conductivity and permittivity, a micro-element of a dielectric is equivalent to a parallel circuit of a capacitor and a resistor. The bipolar charge transport model considers the Schottky injection, charge trappingdetrapping, charge drift-diffusion and charge recombination processes, the following microscopic transport parameters, e.g. carrier mobility, diffusion coefficient, recombination coefficient etc., can be defined flexibly as required [9][10][11][12][13][14].
Although the Maxwell-Wagner model and bipolar charge transport model both aim at describing the charge transport behaviours in dielectrics, they are suitable for different occasions. The Maxwell-Wagner model has done very well in calculating the polarisation current and electric field distribution in various insulation systems, but it is too macroscopic to describe the microscopic behaviours of space charges. So far, the bipolar charge transport model is considered as a preferable method to simulate the space charge accumulation in sheet-form dielectric samples. Although the simulated space charge distribution patterns are similar to those of measurements, there are huge deviations in the values of space charge densities [13]. Besides, the bipolar charge transport model cannot quantitatively relate the microscopic transport behaviours with the macroscopic conductivities of dielectrics, causing failure in accurately predicting the polarisation current and electric field distribution [14]. To solve the above limitations, we have made some modifications to the bipolar charge transport mode based on a viewpoint that carrier dynamic equilibrium exists in solid dielectrics. In this improved model, named carrier dynamic equilibrium model, a source term due to the existence of the dynamic equilibrium of carrier concentration is introduced. By comparing the results of simulations with those of measurement, the carrier dynamic equilibrium model shows the best accordance with measurement, which proved the validity of this improved model.

Space charge measurement and simulation
Low density polyethylene (LDPE) and ethylene propylene diene monomer (EPDM) are widely used in HVDC cable joints as the major insulation [1,2]. Based on the LDPE/EPDM bi-layer dielectric, we study the polarisation process by the pulsed electro acoustic (PEA) measurement and simulation [15]. The thicknesses of LDPE and EPDM were 100 and 140 µm, respectively. The average value of the applied electric field was −40 kV/mm, and the measurement temperature was 30°C. A one-dimensional simulation model was built by using the finite-element method, as shown in Fig. 1. For comparison, three different models, i.e. the Maxwell-Wagner model, bipolar charge transport model, and carrier dynamic equilibrium model, were used to do simulations, and their detailed descriptions are given in Section 3.

Carrier mobility measurement
The surface potential decay (SPD) method was used to measure the mobility of negative and positive carriers of LDPE and EPDM at 30°C, respectively [16]. First, a corona discharge system was used to charge the LDPE or EPDM sample surface, and the saturated surface potential of the sample was kept at ±5 kV by the grid electrode. Then, an electrostatic voltmeter was used to measure the SPD curve through a probe. Finally, the dV/dt-t curve was plotted in logarithmic form, where two tangent lines meet was defined as the SPD characteristic time, as shown in Fig. 2. The carrier mobility can be calculated by using the following equation: where µ is the carrier mobility, m 2 /(V s); L is the sample thickness, m; t T is the SPD characteristic time, s; V 0 is the initial surface potential of SPD.

Maxwell-Wagner model
Based on the principle of conservation of electrical current, the space charge density in the Maxwell-Wagner model is governed by the following equations: where E is the electric field strength, V/m; σ is the volume conductivity of a dielectric, S/m; ɛ 0 is the permittivity of vacuum, F/m; ɛ r is the relative permittivity of a dielectric; Q is the space charge density, C/m 3 .

Bipolar charge transport model
In solid dielectrics, carriers are governed by a set of self-consistent equations as following [9][10][11][12][13][14]17]. The current continuity equation: where Q t − and Q t + are the trap densities, C/m 3 ; P tr and P de are the charge trapping and detrapping coefficients, 1/s; R is the recombination coefficients, m 3 /(C s).
The charge transport equation where µ − and µ + are the mobility of negative and positive carriers, m 2 /(V s); D − and D + are the diffusion coefficients of negative and positive carriers, m 2 /s; E is the electric field strength, V/m. Poisson's equation where V is the electric potential, V; ɛ 0 is the permittivity of vacuum, F/m; ɛ r is the relative permittivity of a dielectric. The boundary conditions: at the boundaries between the dielectric and the metal electrode, the normal current densities can be described as where α = 1 at the cathode and α = 0 at the anode; J s − and J s + are the Schottky injection current densities, which can be described as following: where A is the Richardson constant, C/(m 2 s K 2 ); φ is the contact potential barrier between the dielectric and the metal electrode, eV; E is the electric field strength, V/m; k B is the Boltzmann constant; T is the Kelvin temperature, K.

Carrier dynamic equilibrium model
Although the bipolar charge transport model can describe detailedly the microscopic behaviours of space charges, there is no equation that constringes the polarisation current of a dielectric to a certain value, i.e. the conduction current. Since the polarisation current of a dielectric always converges to the conduction current in measurement, there must be some kind of dynamic equilibrium process that maintains the charge transport capacity of a dielectric at a certain level. For a dielectric, its charge transport capacity mainly depends on carrier mobility and concentration. Considering that carrier mobility is usually treated as a constant at a fixed temperature and electric field, a viewpoint was proposed in this study that the dynamic equilibrium of carrier concentration should exist in solid dielectrics as well as the semiconductors.
In semiconductors, carriers obey the following equation under thermal equilibrium condition: where n 0 and p 0 are the carrier (electron and hole) concentrations in thermal equilibrium state, m −3 ; n i is the intrinsic carrier concentration of a semiconductor, m −3 ; E g is the energy gap, eV; k is the Boltzmann constant; T is the thermal temperature, K. External factors, such as electric field, optical excitation etc., can disturb this equilibrium, leading to the appearance of nonequilibrium carriers where n and p are the carrier concentrations in the non-equilibrium state; Δn and Δp are the non-equilibrium carrier concentrations, whose values can be positive or negative.
To reach another equilibrium state, the non-equilibrium carriers will decline in the negative exponent form where τ is the lifetime of the non-equilibrium carriers, which can be used to assess the speed that the carriers go from the nonequilibrium state to the equilibrium state. The concentration change of the non-equilibrium carriers per unit time is defined as the following: In solid dielectrics, similar carrier dynamic equilibrium can be either the energy band excitation or the impurity ionizationrecombination process. The intrinsic carriers in a solid dielectric can be electrons and holes produced by band excitation or ions produced by impurity ionisation, which obey the following equation [18][19][20]: where q f − and q f + are the charge densities of mobile carriers, C/m 3 ; c 0 is a constant coefficient, C 2 /m −6 ; E q can be regarded as the activation energy of carriers, eV; q i is the charge density of the intrinsic carriers, which can be approximately obtained by using the following equation. In this way, a linkage is established between the macroscopic and microscopic parameters where σ is the measured conductivity of a dielectric, S/m; µ − and µ + are the mobility of positive and negative carriers, m 2 /(V s). Based on the carrier dynamic equilibrium theory, we have made some modifications to the source terms of the bipolar charge transport model by introducing a source term of carrier dynamic equilibrium. The following shows the modified source terms of the carrier dynamic equilibrium model. Since the added source term can indicate both the generation and recombination processes of the mobile carriers, the original recombination term of mobile where S is the added source term of carrier dynamic equilibrium, the positive and negative values represent the mobile carrier generation and recombination processes. Fig. 3 shows a diagram of the charge transport behaviours in LDPE/EPDM bi-layer dielectric based on the carrier dynamic equilibrium model. To simulate the blocking effects of the dielectric/dielectric and dielectric/metal interfaces to mobile carriers, the trapped energy in the interface regions was set to be higher than that in dielectric bulks due to the existence of surface states [21]. The trap energy distribution was assumed to decrease linearly from the dielectric surface to the dielectric bulk in the interface regions [22].
As start conditions, the concentrations of free carriers are in a thermal equilibrium state, the charge trapping and de-trapping processes are also in an equilibrium state. The concentrations of free and trapped carriers at the time t = 0 can be assumed as Table 1 lists some parameters of LDPE and EPDM used in simulations. Fig. 4 presents the measured volume conductivities of LDPE and EPDM by the three-electrode method [23], the volume conductivities of LDPE and EPDM are both electric-field dependent [24].

Effects of carrier mobility
Carrier mobility and concentration are both key parameters in judging the charge transport capability of a dielectric. In this section, we are going to discuss the determinations of the carrier mobility and intrinsic carrier concentration in simulations. In this section, the non-equilibrium carrier lifetime τ was assumed to be 0.1 s, which is larger than that of semiconductors [25]. From (23), one can see that the intrinsic carrier concentration of a dielectric is only dependent on temperature. Since the measured volume conductivities of LDPE and EPDM both increase with electric field strength, the carrier mobility of LDPE and EPDM should be electric-field dependent. In Section 2, we have introduced the SPD method to measure the carrier mobility of LDPE and EPDM, measurement results of the positive and negative carriers are shown in Fig. 5a. Since the electric field strength inside a sample declines from 10 kV/mm to almost zero during the measurement, the carrier mobility should be lower than that under 10 kV/mm but higher than that under 0 kV/mm where µ mea is the measured carrier mobility, µ(0) and µ(10) are the carrier mobility under electric field strength of 0 and 10 kV/mm, respectively. By combining (24) and (30), the carrier mobility scopes of LDPE and EPDM can be determined as shown in Figs. 5b and c

Fig. 3 Diagram of the charge transport behaviours in LDPE/EPDM bi-layer dielectric
where σ(E) is the electric-field-dependent volume conductivity; σ(0) and σ (10) are the volume conductivities under electric field strength of 0 and 10 kV/mm; µ(E) is the electric-field-dependent carrier mobility. By setting the mobility of negative and positive carriers to their upper or lower limits, we obtain the scopes of the intrinsic carrier concentrations in LDPE and EPDM materials, as shown in Fig. 5d.
To explore the influence of carrier mobility on the simulation results of the carrier dynamic equilibrium model, Fig. 6 presents the simulated polarisation current, steady-state space charge and electric field distributions in LDPE/EPDM bi-layer dielectric when assigning the upper limit, lower limit, and average value to the carrier mobility, respectively. All the polarisation currents decline to their steady-states in 1800 s, and a slight increment of conduction current with increasing carrier mobility is observed. Although the volume conductivities of LDPE and EPDM are kept invariable, higher carrier mobility can accelerate the departure process of the injected carriers from electrodes, which increase the conduction current density and suppress the homo-charge accumulation in the electrode regions. By integrating the space charge density in the interface region, the total interface charge density of negative polarity is found to be ∼1.06 mC/m 2 and barely changes with the carrier mobility. To evaluate the electric field distribution, the distortion rate is defined in the following equation: where f is the electric field distortion rate; L is the sample thickness, m; E ave is the average electric field strength, V/m. The distortion rates of steady-state electric field distributions are ∼0.154, no obvious change of the electric field distribution can be observed while increasing the carrier mobility. Since the change of carrier mobility does not have a very significant influence on the simulation results, the carrier mobility is defined as an average value of its upper and lower limits in the following sections.

Effects of non-equilibrium carrier lifetime
In this study, the carrier dynamic equilibrium source is introduced to modify the charge transport model, and the non-equilibrium carrier lifetime is thus a key parameter to describe the dynamic behaviours of carrier concentrations. Since there is no effective method to measure the non-equilibrium carrier lifetime of dielectrics, in this section, we are going to discuss the influence of the non-equilibrium carrier lifetime on the simulation results based on four hypothetical values of 0.001, 0.1, 10, and 1000 s. As shown in Fig. 7, when the non-equilibrium carrier lifetime is below 10 s, no obvious difference can be distinguished among the simulation results. However, when the non-equilibrium carrier lifetime reaches 1000 s, the polarisation current shows a much slower decline trend and does not reach the steady state after 1800 s. Moreover, the interface charge density of negative polarity is reduced when setting the non-equilibrium carrier lifetime up to 1000 s, so does the electric field distortion caused by interface charge accumulation.  Fig. 8 shows the dynamic behaviours of space charge and electric field distributions in the polarisation process of LDPE/EPDM bilayer dielectric (assuming the non-equilibrium carrier lifetime τ = 0.1 s, the carrier mobility is defined as an average value of its upper and lower limits). In the electrode regions, the densities of homo-charges due to the Schottky injection effect increase gradually over time. Meanwhile, negative charge accumulation at the interface of LDPE and EPDM causes severe electric field distortion in the bi-layer dielectric. During the polarisation process, the electric field strength in EPDM becomes increasingly higher than that in LDPE, the electric field distortion rate increases with time and finally reaches a steady value of ∼0.154 after 1800 s. Since the trapping coefficient of mobile carriers is higher than the de-trapping coefficient of trapped carriers, most of the space charges are found in trap sites with and without electric field application. Fig. 9 shows a comparison among the results of measurement and different simulation models. The Maxwell-Wagner model performs well in predicting the conduction current and electric field distribution in LDPE/EPDM bi-layer dielectric, but the interface charge turns out to accumulate at an ideal interface without thickness, which is quite different from the measurement.

Comparison between simulation and measurement
The bipolar charge transport model has a similar space charge distribution pattern to that of measurement, but the total interface charge density is obviously lower, so does the electric field distortion. Besides, the polarisation current of the bipolar charge transport model keeps going down with time and deviates further and further from the measured conduction current. By introducing the carrier dynamic equilibrium source, the improved model has a better coincidence with measurement than both the Maxwell-Wagner model and the bipolar charge transport model. The simulated conduction current density, interface charge density, and electric field distortion are all very close to the measurement data, which proves the validity of the existence of carrier dynamic equilibrium in dielectrics.

Conclusions
The carrier dynamic equilibrium model was proposed in this study to simulate the polarisation process of LDPE/EPDM bi-layer dielectric, based on a viewpoint that carrier dynamic equilibrium exists in solid dielectrics. A source term of carrier dynamic equilibrium was introduced to establish a linkage between the macroscopic conductivities and microscopic charge transport behaviours of dielectrics. In spite of its crudeness, the simulated conduction current density, interface charge density, and electric field distortion of the carrier dynamic equilibrium model are all closer to the measurement data, compared with those of the 'macroscopic' Maxwell-Wagner model and the 'microscopic' bipolar charge transport model. This model can be used to calculate the space charge and electric field distributions in various power equipment, which will help design and optimise the HVDC insulation system.