A Finite Element Analysis Model for Partial Discharges in Silicone Gel under a High Slew Rate, High-Frequency Square Wave Voltage in Low-Pressure Conditions

Wide bandgap (WBG) devices made from materials such as SiC, GaN, Ga2O3 and diamond, which can tolerate higher voltages and currents compared to silicon-based devices, are the most promising approach for reducing the size and weight of power management and conversion systems. Silicone gel, which is the existing commercial option for encapsulation of power modules, is susceptible to partial discharges (PDs). PDs often occur in air-filled cavities located in high electric field regions around the sharp edges of metallization in the gel. This study focuses on the modeling of PD phenomenon in an air filled-cavity in silicone gel for the combination of (1) a fast, high-frequency square wave voltage and (2) low-pressure conditions. The low-pressure condition is common in the aviation industry where pressure can go as low as 4 psi. To integrate the pressure impact into PD model, in the first place, the model parameters are adjusted with the experimental results reported in the literature and in the second place, the dependencies of various PD characteristics such as dielectric constant and inception electric field on pressure are examined. Finally, the reflections of these changes in PD intensity, duration and inception time are investigated. The results imply that the low pressure at high altitudes can considerably affect the PD inception and extinction criterion, also the transient state conditions during PD events. These changes result in the prolongation of PD events and more intense ones. As the PD model is strongly dependent upon the accurate estimation electric field estimation of the system, a finite-element analysis (FEA) model developed in COMSOL Multiphysics linked with MATLAB is employed that numerically calculates the electric field distribution.


Introduction
In the commercial aircraft segment, the annual growth in global air traffic passenger demand is expected at 4.7% until 2028 [1] with passenger numbers expected to double in the next 20 years until around 2040 [2]. However, the commercial aviation industry will have to face both economic and environmental issues. In 2017, the U.S. commercial aviation consumed the equivalent of about 16% by volume of petroleum (crude oil and products) imports [3]. In the same year, air travel accounted for about 174.8 million metric tons of CO 2 equivalent emissions in the U.S. or about 2.6% of domestic greenhouse gas emissions [4]. To decarbonize and reduce energy consumption, enabling technologies for electrified aircraft propulsion have been targeted. In this regard, wide bandgap (WBG)-based drives are a promising solution to achieve high power density and efficiency simultaneously for all-electric powertrain. as well as COMSOL Multiphysics and MATLAB, this study takes advantage of FEA to investigate the PD phenomenon.
In Reference [27], the detrimental impact of slew rate was studied and the results showed that high slew rate square voltages substantially increased the intensity of PDs within the insulating medium and changed the PD behavior over time, that is, the repetition rate and the times at which discharges occur. It has also been shown in References [28,29] that, although at a higher frequency the PD intensity is lower, the higher repetition rate causes rapid degradation of insulation. Thus, at higher frequencies, the lifetime of insulation is reduced. In Reference [30], the simulation of partial discharge is performed for square wave voltage, however, the applied voltage frequency is only 200 Hz. In Reference [31], the results of tests on single-point contact crossed pairs are reported to assess the impact of rise-time (200 ns to 400 µs) on PD intensity and frequency spectrum. The results indicated that high-frequency components proliferate in the PD spectrum of energy at low slew rates of square wave voltage. The authors further extended their studies toward the impact of high frequency (up to 5 kHz) voltage pulses on the motor insulation life [29] and it is shown that the physical and chemical deterioration rate of insulation is influenced at higher frequencies. While the reported experimental results provided invaluable clues on the impact of fast-rise, high-frequency square wave voltages, no study has been performed on the modeling of PD phenomenon under this type of voltage. To address this gap, a FEA method was developed in Reference [28] for modeling PD characteristics under fast, repetitive voltage pulses.
Aside from the impact of the applied voltage characteristics, environmental conditions can also change the way the PD phenomenon is looked at today. For instance, an aircraft can be at a pressure as low as 4 psi when flying at cruising altitude where the impact of pressure on PD occurrence has not been sufficiently studied so far. To the best of our knowledge, little research appears on the effect of fast, repetitive voltage pulses at low-pressure levels on PD behavior. For example, in Reference [32] test results under square wave voltages with frequencies in 5-200 kHz range at pressure levels between 20 and 100 kPa implied that the PD inception voltage is constantly decreasing with pressure.
This study focuses on the effects that low-pressure conditions have on different model parameters and reflects on how these changes affect PD characteristics. The work was done under high frequency, high slew rate voltages to accurately model the escalated tension that the insulation system undergoes. This paper takes advantage of the FEA model for the above goal. The advantages of FEA over other models are its ability to apply to sophisticated geometries as well as its ability to dynamically monitor changes in the values of electrical properties such as electric field, current density, field displacement and so forth [33].
In Section 2, the effect of pressure on PD inception, extinction and dielectric constant is analyzed. In Section 3, PD modeling details are discussed. Section 4 is dedicated to a case study where the proposed modeling is applied and the results are discussed. Section 5 gives conclusions.

PD Inception
A discharge occurs when a collision takes place between free electrons and other atoms/molecules, giving rise to an electron avalanche. The existence of at least one free electron is necessary for ionization to occur. Besides, the electric field should be high enough to energize the electrons when colliding with other particles.
In Reference [21], a critical number of electrons at the avalanche head that could guarantee the self-propagation of the ionization process was proposed. For an ellipsoidal void, the mentioned condition known as streamer inception criterion can be described as: Energies 2020, 13, 2152 4 of 11 In the above expression, (E/p) cr , B and n are gas parameters. In air, their values are 24.2 V Pa −1 m −1 , 8.6 Pa 0.5 m 0.5 and 0.5, respectively. The pressure of air is denoted as p, while l represents the ellipsoidal cavity diameter in the direction of electric field.

PD Extinction
Discharge occurs through an increase in the cavity conductivity from an initial value (approximately zero) to a higher value (σ cav, max ). An increase in conductivity results in a decrease in the electric field intensity inside the cavity. Conductivity remains at a high value until the electric field recedes to a certain magnitude, called the Extinction Electric Field (E ext ). According to Gutfleisch and Niemeyer [34,35], the electric field at which PD is quenched can be described as a function of the critical electric field: In the above expression, γ is a dimensionless factor varying with the change of voltage polarity. γ is experimentally adjusted for the normal temperature and pressure (NTP) conditions (p 0 , T 0 ).

Dielectric Constant (Relative Permittivity)
As this study strives to model partial discharge throughout changes in pressure magnitude, it is crucial to take into account the effect of pressure on relative permittivity. The reason for this is that the dielectric constant can change the magnitude of the electric field according to the Maxwell equation and PD modeling is strongly dependent upon the accurate estimation of electric field distribution.
Owen and Brinkley suggested that the relative permittivity of a liquid dielectric can be given by Reference [36]: where p 0 and p 0 denote the relative permittivity and pressure, respectively, under NTP conditions. Similarly, p stands for the relative permittivity at pressure p. A and B are constants of Tait expression related to the material and must be experimentally determined. Thus, in this study, approximate values are derived from Reference [37]. This expression takes only pressure into account. However, the author mentions that parameter A is independent of temperature and B is often decreasing with temperature.
Since silicone gel has properties of liquid dielectrics in terms of viscosity, in this paper, we assume that (3) can be used for silicone gel as well. In this regard, Figure 1 shows the changes in silicone gel relative permittivity as a function of pressure. Generally, the figure implies that the relative permittivity has a direct relationship with pressure; however, the rate of change declines as pressure is reduced and r has little variations for p < 1 atm.

PD Extinction
Discharge occurs through an increase in the cavity conductivity from an initial value (approximately zero) to a higher value ( , ). An increase in conductivity results in a decrease in the electric field intensity inside the cavity. Conductivity remains at a high value until the electric field recedes to a certain magnitude, called the Extinction Electric Field ( ). According to Gutfleisch and Niemeyer [34,35], the electric field at which PD is quenched can be described as a function of the critical electric field: In the above expression, is a dimensionless factor varying with the change of voltage polarity. is experimentally adjusted for the normal temperature and pressure (NTP) conditions ( , ).

Dielectric Constant (Relative Permittivity)
As this study strives to model partial discharge throughout changes in pressure magnitude, it is crucial to take into account the effect of pressure on relative permittivity. The reason for this is that the dielectric constant can change the magnitude of the electric field according to the Maxwell equation and PD modeling is strongly dependent upon the accurate estimation of electric field distribution.
Owen and Brinkley suggested that the relative permittivity of a liquid dielectric can be given by Reference [36]: where and denote the relative permittivity and pressure, respectively, under NTP conditions. Similarly, stands for the relative permittivity at pressure . and are constants of Tait expression related to the material and must be experimentally determined. Thus, in this study, approximate values are derived from Reference [37]. This expression takes only pressure into account. However, the author mentions that parameter is independent of temperature and is often decreasing with temperature.
Since silicone gel has properties of liquid dielectrics in terms of viscosity, in this paper, we assume that (3) can be used for silicone gel as well. In this regard, Figure 1 shows the changes in silicone gel relative permittivity as a function of pressure. Generally, the figure implies that the relative permittivity has a direct relationship with pressure; however, the rate of change declines as pressure is reduced and has little variations for < 1 atm.

PD Modeling
As disclosed earlier, the electric field intensity plays a major role in different sections of PD modeling. Therefore, an accurate estimation of the electric field distribution is crucial to the whole process. In this regard, the current study employs FEA to utilize the capabilities of the finite-element method to accurately estimate the electric field distribution within the air filled-cavity, silicone gel system.
In this work, COMSOL Multiphysics is used to implement FEA. Therefore, the electric current (ec) interface of COMSOL Multiphysics is used in a time-dependent mode. The boundary conditions include the setting of the high-voltage electrode boundary to the applied voltage waveform and the ground electrode boundary to zero voltage. Another condition is that no current flows into the boundary that is not included in the aforementioned boundary conditions. LiveLink TM for MATLAB is also used along with COMSOL to model the other parts of the algorithm such as checking conditionals within the PD model and calculating the true and apparent charge magnitudes. Moreover, the interface also plays a role in the analysis of the PD sensitivity to the affecting parameters.
To calculate the true and apparent charge magnitudes, the algorithm uses one of the outputs of COMSOL, which is the current density flowing through the cavity wall and ground electrode. Then, by integrating the current density over time, starting from inception and going to extinction, the charge magnitudes are obtained.
The PD modeling process is shown in Figure 2. Starting from the beginning of the period, the PD prerequisites are examined at each time step (∆t H ). Before the occurrence of a PD event, the cavity conductivity is set to its initial value (σ cav,0 ). The electric field distribution is obtained for each time step by FEA model to evaluate the PD occurrence requirements (i.e., streamer inception criterion and the availability of initial free electron).

PD Modeling
As disclosed earlier, the electric field intensity plays a major role in different sections of PD modeling. Therefore, an accurate estimation of the electric field distribution is crucial to the whole process. In this regard, the current study employs FEA to utilize the capabilities of the finite-element method to accurately estimate the electric field distribution within the air filled-cavity, silicone gel system.
In this work, COMSOL Multiphysics is used to implement FEA. Therefore, the electric current (ec) interface of COMSOL Multiphysics is used in a time-dependent mode. The boundary conditions include the setting of the high-voltage electrode boundary to the applied voltage waveform and the ground electrode boundary to zero voltage. Another condition is that no current flows into the boundary that is not included in the aforementioned boundary conditions. LiveLink TM for MATLAB is also used along with COMSOL to model the other parts of the algorithm such as checking conditionals within the PD model and calculating the true and apparent charge magnitudes. Moreover, the interface also plays a role in the analysis of the PD sensitivity to the affecting parameters.
To calculate the true and apparent charge magnitudes, the algorithm uses one of the outputs of COMSOL, which is the current density flowing through the cavity wall and ground electrode. Then, by integrating the current density over time, starting from inception and going to extinction, the charge magnitudes are obtained.
The PD modeling process is shown in Figure 2. Starting from the beginning of the period, the PD prerequisites are examined at each time step (Δ ). Before the occurrence of a PD event, the cavity conductivity is set to its initial value ( , ). The electric field distribution is obtained for each time step by FEA model to evaluate the PD occurrence requirements (i.e., streamer inception criterion and the availability of initial free electron).
If any of the requirements are not met, the process adds a time step (Δ ) to the time until reaching a moment that both PD conditions are met or the process reaches the end of the time. For the times that the occurrence of PD is confirmed, the conductivity of the cavity is enhanced as explained earlier. In the next time steps (Δ ), the PD extinction is looked for after running FEA.
The time steps referred to during this algorithm vary according to the state of the process. As the changes during PD are in order of nanoseconds, the time step during PD (Δ ) is much smaller than the time step during no-PD (Δ ) period.

Results and Key Challenges
The geometry that was examined for the modeling and simulations is shown in Figure 3a and consists of two spherical electrodes; one is the high-voltage electrode while the other is grounded, both made of copper. The two spheres are situated within a cylindrical block of silicone gel ( = 2.7 at NTP) with a spherical air-filled cavity inside. The geometric parameters of the void-dielectric set are shown in Table 1. This geometry with silicone gel as its dielectric material was first proposed and If any of the requirements are not met, the process adds a time step (∆t H ) to the time until reaching a moment that both PD conditions are met or the process reaches the end of the time. For the times that the occurrence of PD is confirmed, the conductivity of the cavity is enhanced as explained earlier.
In the next time steps (∆t L ), the PD extinction is looked for after running FEA.
The time steps referred to during this algorithm vary according to the state of the process. As the changes during PD are in order of nanoseconds, the time step during PD (∆t L ) is much smaller than the time step during no-PD (∆t H ) period.

Results and Key Challenges
The geometry that was examined for the modeling and simulations is shown in Figure 3a and consists of two spherical electrodes; one is the high-voltage electrode while the other is grounded, Energies 2020, 13, 2152 6 of 11 both made of copper. The two spheres are situated within a cylindrical block of silicone gel (ε r = 2.7 at NTP) with a spherical air-filled cavity inside. The geometric parameters of the void-dielectric set are shown in Table 1. This geometry with silicone gel as its dielectric material was first proposed and tested in Reference [38]. The authors in their previous work [28], estimated the parameters associated with PD model such that it reproduces the same results as in Reference [38] and chose it as a solid groundwork for further investigations. Due to the symmetry of this configuration, it can be also demonstrated in a 2D-axisymmetrical format as shown in Figure 3b. Using the 2D demonstration instead of a 3D figure significantly reduces FEA calculation burden.   The properties of the unipolar square wave voltage are shown in Table 2.  Figure 4 demonstrates that and have opposite trends with pressure variation. is a decreasing function of pressure, while is an increasing function. As will be shown later, these two trends play a significant role in PD characteristics such as PD inception and extinction times, intensity and so forth.
The rest of this section focuses on the variations of several PD characteristics with pressure. Simulations are performed for 15 pressure values which are shown in red points in Figures 5-7. Then, the trends of the variations denoted as a fitted curve in Figures 5-7 are derived over a continuous range of pressures using smoothing spline interpolation technique.  The properties of the unipolar square wave voltage are shown in Table 2.  Figure 4 demonstrates that E inc and E ext have opposite trends with pressure variation. E inc is a decreasing function of pressure, while E ext is an increasing function. As will be shown later, these two trends play a significant role in PD characteristics such as PD inception and extinction times, intensity and so forth.   increases, the PD demands more time to catch a higher electric field value. A quick conclusion might be that the later PD occurs, the longer the insulation material can withstand degradation; however, as will be discussed next, a bit later discharges impose more intense ones. Similarly, the extinction time trend can be justified; at lower pressures, the linear function of the extinction field, concerning pressure, trends downward as well. To quench a discharge at lower field magnitudes, more time in the high-conductivity period is needed to reduce the electric field intensity. Thus, like inception time, extinction time is also higher for lower pressures. Since both inception and extinction times increase with pressure reduction, what happens to the duration of PD at lower pressures? Figure 6 indicates that the duration of discharges at lower pressures are higher than NTP conditions. Thus, it can be inferred that, under low-pressure conditions, the prolongation of PDs due to later extinction times is much higher than the shortening of PDs due to later inception times. This difference tends to decline as the pressure goes up.   increases, the PD demands more time to catch a higher electric field value. A quick conclusion might be that the later PD occurs, the longer the insulation material can withstand degradation; however, as will be discussed next, a bit later discharges impose more intense ones. Similarly, the extinction time trend can be justified; at lower pressures, the linear function of the extinction field, concerning pressure, trends downward as well. To quench a discharge at lower field magnitudes, more time in the high-conductivity period is needed to reduce the electric field intensity. Thus, like inception time, extinction time is also higher for lower pressures. Since both inception and extinction times increase with pressure reduction, what happens to the duration of PD at lower pressures? Figure 6 indicates that the duration of discharges at lower pressures are higher than NTP conditions. Thus, it can be inferred that, under low-pressure conditions, the prolongation of PDs due to later extinction times is much higher than the shortening of PDs due to later inception times. This difference tends to decline as the pressure goes up.  Figure 7 shows variations of PD true charge magnitude as a function of pressure. The results affirm that as the pressure is reduced, the PD magnitude increases. There are two different mechanisms that play different roles. One is that, at lower pressures, discharge occurs at higher field magnitudes; according to Ohm's law, at higher field magnitudes, the current density will be higher. ( = ). Since the field displacement has a direct relationship with the true charge magnitude, at a lower permittivity the amount of true charge would be lower. Figure 1 shows that the variations of relative permittivity at low pressures are not as sharp as at high pressures; hence, the overall increased magnitude of true charge is due to the deeper impact of enhancement versus reduction at low pressures.

Conclusions
The growing trend toward the electrification of the aviation industry urges the demand for ensuring the reliability of the insulation system. The utilization of WBG-based power modules can impose further threats to the insulation systems due to their operation at a higher frequency and higher slew rate. Because the pressure at higher altitudes can be extremely low, this study examined the impact of high frequency, high slew rate unipolar square wave voltage under low-pressure conditions. It was shown that low-pressure conditions have a game-changing impact by influencing the PD inception and extinction times, duration and intensity. These changes are the result of changes in the inception and extinction electric fields as well as the relative permittivity of the insulation medium. The results show that despite a slight delay in PD initiation under low-pressure conditions, the true charge magnitude is increased by about 20% and the duration of PD events is nearly doubled at cruise altitude compared to the ground level. As the material nature has a substantial impact on the extent and nature of the  Figure 5 reflects the impact of pressure on the inception time of PDs. Higher inception time values at lower pressures can be explained by the fact that as the pressure decreases, E inc goes up. As E inc increases, the PD demands more time to catch a higher electric field value. A quick conclusion might be that the later PD occurs, the longer the insulation material can withstand degradation; however, as will be discussed next, a bit later discharges impose more intense ones.
Similarly, the extinction time trend can be justified; at lower pressures, the linear function of the extinction field, concerning pressure, trends downward as well. To quench a discharge at lower field magnitudes, more time in the high-conductivity period is needed to reduce the electric field intensity. Thus, like inception time, extinction time is also higher for lower pressures. Since both inception and extinction times increase with pressure reduction, what happens to the duration of PD at lower pressures? Figure 6 indicates that the duration of discharges at lower pressures are higher than NTP conditions. Thus, it can be inferred that, under low-pressure conditions, the prolongation of PDs due to later extinction times is much higher than the shortening of PDs due to later inception times. This difference tends to decline as the pressure goes up. Figure 7 shows variations of PD true charge magnitude as a function of pressure. The results affirm that as the pressure is reduced, the PD magnitude increases. There are two different mechanisms that play different roles. One is that, at lower pressures, discharge occurs at higher field magnitudes; according to Ohm's law, at higher field magnitudes, the current density will be higher. A higher current density then leads to a higher discharge magnitude. On the other hand, the relative permittivity declines at lower pressures, which means that the field displacement would be lower (D = E). Since the field displacement has a direct relationship with the true charge magnitude, at a lower permittivity the amount of true charge would be lower. Figure 1 shows that the variations of relative permittivity at low pressures are not as sharp as at high pressures; hence, the overall increased magnitude of true charge is due to the deeper impact of E inc enhancement versus r reduction at low pressures.

Conclusions
The growing trend toward the electrification of the aviation industry urges the demand for ensuring the reliability of the insulation system. The utilization of WBG-based power modules can impose further threats to the insulation systems due to their operation at a higher frequency and higher slew rate. Because the pressure at higher altitudes can be extremely low, this study examined the impact of high frequency, high slew rate unipolar square wave voltage under low-pressure conditions. It was shown that low-pressure conditions have a game-changing impact by influencing the PD inception and extinction times, duration and intensity. These changes are the result of changes in the inception and extinction electric fields as well as the relative permittivity of the insulation medium. The results show that despite a slight delay in PD initiation under low-pressure conditions, the true charge magnitude is increased by about 20% and the duration of PD events is nearly doubled at cruise altitude compared Energies 2020, 13, 2152 9 of 11 to the ground level. As the material nature has a substantial impact on the extent and nature of the aforementioned changes, more research on different insulating materials can lead to the development of a more reliable insulation system under extremely low-pressure conditions.