3.1. Simulation model construction

2.4. Niemeyer's model

Niemeyer considered PD within the cavity as a streamer-type discharge, because only this type could be detected and has engineering significance [9]. After analyzing the physical processes

Eq. (5) is actually the well-known critical avalanche criterion, in which α, the function of electric field, indicates the effective ionization coefficient, Kcr the logarithm of a critical number of electrons that has to accumulate in the avalanche head to make the avalanche self-propagating by its own space charge field, and xcr the distance within α which exceeds zero. In terms of it, the inception field of PD occurrence could be obtained. Eq. (6) simply describes the streamer propagation, where Ech is the electric field in the discharge channel, Ures the residual voltage instantaneously after discharge, and lstr the distance to which streamer could propagate. Eq. (7) establishes the relationship between physical charges and potential difference before and after a

Based on the model, Niemeyer simulated PD behaviors within a spherical cavity by considering the stochastic supply of free electrons, which agreed with experimental data qualitatively and quantitatively although there was a slight disagreement in the phase and magnitude distributions of PD. However, there is a significant shortcoming that the electric field distribution was assumed to be uniform within the cavity. Considering this point, Illias developed the simulation model in which the deployed charges were not uniform and Poisson's equation

In terms of physical processes, a cavity PD is similar to the filamentary dielectric barrier discharge (DBD) [25]. As for the latter, fluid equations are widely used to simulate gas discharge process [26, 27], which describe the impact ionization, charge drift, diffusion, recombination, and some secondary effects. In recent years, several researchers employed them to simulate the PD occurring in a cavity [18–20]. For example, Novak and Bartnikas established a two-dimensional breakdown model based on the continuity equations for electrons and ions to examine the influence of surface charges upon the partial discharge behavior [19]. In terms of it, the evolution of electric field and charge concentration distribution within the cavity during

However, the behaviors of single PD could not represent that of continuous PDs due to the memory effect. On one hand, residual charges generated by previous discharge land on the cavity surface and affect the electric field distribution within the cavity, leading to the change of subsequent PD characters. On the other hand, the accumulated surface charges may provide

α½ � E xð Þ dx ≥ Kcr (5)

ΔUres ≈ Echlstr (6)

q ¼ �gπε0lΔUPD (7)

of PD, he proposed several equations to describe PD, as follows:

114 Plasma Science and Technology - Basic Fundamentals and Modern Applications

xðcr

0

PD, in which g is a dimensionless proportionality factor and l the cavity scale.

the discharge process was obtained, as well as the discharge current pulse.

was employed to calculate PD parameters [16, 17].

2.5. Plasma model

Because sandwich-type samples are widely used in the experimental researches on PD, a cylindrical cavity with a diameter of 2 mm and a height of 0.25 mm is employed in our simulation model, as in Figure 3. The cavity, full of atmospheric pressure air, is embedded within the solid dielectric, of which the relative permittivity equals 2.3. The thickness of dielectric barriers is set to be identical to the cavity height. Although during the discharge process, the temperature of cavity may slightly increase due to the joule heating from discharges, the temperature variation is neglected in our model, which means that the pressure in the cavity keeps unchanged.

The streamer development is quantitatively described by fluid equations, as follows:

$$\frac{\partial N\_{\rm e}}{\partial t} = N\_{\rm e} \alpha |\mathbf{W\_{e}}| - N\_{\rm e} \eta |\mathbf{W\_{e}}| - N\_{\rm e} N\_{\rm p} \beta - \nabla \cdot (N\_{\rm e} \mathbf{W\_{e}} - D \nabla N\_{\rm e}) \tag{8}$$

$$\frac{\partial N\_{\rm P}}{\partial t} = N\_{\rm c}a|\mathbf{W\_{e}}| - N\_{\rm c}N\_{\rm p}\beta - N\_{\rm n}N\_{\rm p}\beta - \nabla \cdot \left(N\_{\rm p}\mathbf{W\_{p}}\right) \tag{9}$$

Figure 3. Configuration of simulation model.

$$\frac{\partial \mathbf{N\_{n}}}{\partial t} = \mathbf{N\_{e}}\eta|\mathbf{W\_{e}}| - \mathbf{N\_{n}}\mathbf{N\_{p}}\beta - \nabla \cdot (\mathbf{N\_{n}}\mathbf{W\_{n}}) \tag{10}$$

At the upper and lower surfaces of cavity, the boundary conditions for Poisson's equation are

� <sup>ε</sup>rε0E<sup>z</sup> <sup>d</sup><sup>þ</sup>

g

both sides of the upper surface, while Ez 0� ð Þ and Ez 0<sup>þ</sup> ð Þ represent the z-component of electric field at both sides of the lower surface, σ<sup>u</sup> and σ<sup>d</sup> denote the surface charge density at the

An initial electron-positive ion pair with a concentration of 10<sup>13</sup> cm�<sup>3</sup> is placed near the upper or lower surface to induce the streamer and avoid Townsend phase of gas discharge [28]. It should be noted that this assumption differs from the consideration of free electrons, which will be described in the later text. During the streamer development, charge concentration varies quickly, and an area with a steep concentration gradient appears at the head of the streamer. Meanwhile, the value of charge concentration should maintain positive, which cannot be guaranteed by the traditional finite difference method. So, the flux-corrected transport (FCT) algorithm is used to solve the convection term of charge continuity equations to

In general, the time step for FCT is chosen based on the electron drift velocity, however, which may not apply to the circumstance in our simulation model. It is because apart from the streamer development, its extinguishment process also needs to be obtained which is responsible for the accumulation of electrons and ions. However, the drift velocity of electrons is about 100 higher than that of ions, and the choice of time step must lead to the large increase of calculation consumption at the later stage of discharge when ion drift dominates. Instead, if it is chosen based on the ion drift velocity, the accuracy of the calculation cannot be guaranteed at the initial stage of discharge. Therefore, as a compromise, the time step is set according to whether there are any electrons within the cavity volume. In detail, during the initial stage of streamer development, it is determined by the electron drift velocity. After electrons completely accumulate at the interface, it depends on the drift velocity of a positive ion or a negative one (both are the same). The expression

<sup>Δ</sup>te,<sup>p</sup> <sup>¼</sup> <sup>0</sup>:<sup>1</sup> <sup>Δ</sup><sup>z</sup>

where <sup>Δ</sup><sup>t</sup> is the time step, <sup>Δ</sup><sup>z</sup> the grid length along <sup>z</sup>-direction, and j j <sup>W</sup> max the maximum value

According to Pedersen's model, the apparent charges are determined by charge transportation within the cavity, which could be detected by pulse current method. However, due to the effect of dielectric barriers, the pulse obtained at the external circuit may not reflect the streamer propagation. So, we use Sato's equation to calculate the current due to free charge movement [34], as follows:

We,<sup>p</sup> 

g

<sup>¼</sup> <sup>σ</sup><sup>u</sup> (13)

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max (15)

indicate the <sup>z</sup>-component of electric field at

Numerical Modeling of Partial Discharge Development Process

εrε0E<sup>z</sup> 0� ð Þ� ε0E<sup>z</sup> 0<sup>þ</sup> ð Þ¼ σ<sup>d</sup> (14)

ε0E<sup>z</sup> d� g

g

overcome the two problems [31–33], which is listed in Appendix B.

and Ez <sup>d</sup><sup>þ</sup>

where d<sup>g</sup> is the cavity height, Ez d�

upper and lower surfaces.

for the time step is

of charge drift velocity within the cavity.

where N indicates the bulk charge concentration within the cavity, e, p, and n the symbols for electron, positive ion, and negative ion, respectively, t discharge time, α, η, β, and D denote the ionization, attachment, recombination, and electron diffusion coefficients, respectively, and W the drift velocity. Eqs. (8)–(10) reflect the transportation processes of electrons, positive and negative ions, which includes impact ionization, drift, diffusion, attachment, and recombination. However, the secondary processes, for example, photoionization, are neglected due to two reasons: (1) photoionization is crucial to the streamer development in long gaps but not so important for short gaps [28] and (2) the calculation of the secondary effect is extremely complicated, especially for the photoionization [29], which would bring about great difficulties of the PD sequence simulation. The detailed expressions of the above transport parameters come from Morrow's paper [30], and we list them in Appendix A.

After the streamer arrives at the interface between the cavity and the dielectric, the charges will accumulate on the dielectric surface. We use the following equation to describe the transition from volume charges to surface charges:

$$
\sigma \Delta \mathbf{S} = \left( \mathbf{N\_p} - \mathbf{N\_e} - \mathbf{N\_n} \right) e \Delta V \tag{11}
$$

where ΔS and ΔV represent the area and volume of unit grid after meshing, respectively. Surface charge distribution is assumed to keep unchanged during the discharge process.

During the streamer development, the influence of space charges on the electric field should not be neglected, so Poisson's equation is employed to obtain the electric field within the cavity:

$$\nabla^2 \varphi = -\frac{e}{\varepsilon\_\text{tr}\varepsilon\_0} \left( N\_\text{p} - N\_\text{e} - N\_\text{n} \right) \tag{12}$$

At the upper and lower surfaces of cavity, the boundary conditions for Poisson's equation are

$$
\varepsilon\_0 E\_\mathbf{z} \left( d\_\mathbf{g}^+ \right) - \varepsilon\_\mathbf{r} \varepsilon\_0 E\_\mathbf{z} \left( d\_\mathbf{g}^+ \right) = \sigma\_\mathbf{u} \tag{13}
$$

$$
\varepsilon\_\mathbf{r} \varepsilon\_0 E\_\mathbf{z}(\mathbf{0}^-) - \varepsilon\_0 E\_\mathbf{z}(\mathbf{0}^+) = \sigma\_\mathbf{d} \tag{14}
$$

where d<sup>g</sup> is the cavity height, Ez d� g and Ez <sup>d</sup><sup>þ</sup> g indicate the <sup>z</sup>-component of electric field at both sides of the upper surface, while Ez 0� ð Þ and Ez 0<sup>þ</sup> ð Þ represent the z-component of electric field at both sides of the lower surface, σ<sup>u</sup> and σ<sup>d</sup> denote the surface charge density at the upper and lower surfaces.

An initial electron-positive ion pair with a concentration of 10<sup>13</sup> cm�<sup>3</sup> is placed near the upper or lower surface to induce the streamer and avoid Townsend phase of gas discharge [28]. It should be noted that this assumption differs from the consideration of free electrons, which will be described in the later text. During the streamer development, charge concentration varies quickly, and an area with a steep concentration gradient appears at the head of the streamer. Meanwhile, the value of charge concentration should maintain positive, which cannot be guaranteed by the traditional finite difference method. So, the flux-corrected transport (FCT) algorithm is used to solve the convection term of charge continuity equations to overcome the two problems [31–33], which is listed in Appendix B.

∂N<sup>n</sup>

116 Plasma Science and Technology - Basic Fundamentals and Modern Applications

Figure 3. Configuration of simulation model.

come from Morrow's paper [30], and we list them in Appendix A.

∇2

<sup>φ</sup> ¼ � <sup>e</sup> εrε<sup>0</sup>

from volume charges to surface charges:

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>N</sup>eηj j <sup>W</sup><sup>e</sup> � <sup>N</sup>nNp<sup>β</sup> � <sup>∇</sup> � ð Þ <sup>N</sup>nW<sup>n</sup> (10)

eΔV (11)

(12)

where N indicates the bulk charge concentration within the cavity, e, p, and n the symbols for electron, positive ion, and negative ion, respectively, t discharge time, α, η, β, and D denote the ionization, attachment, recombination, and electron diffusion coefficients, respectively, and W the drift velocity. Eqs. (8)–(10) reflect the transportation processes of electrons, positive and negative ions, which includes impact ionization, drift, diffusion, attachment, and recombination. However, the secondary processes, for example, photoionization, are neglected due to two reasons: (1) photoionization is crucial to the streamer development in long gaps but not so important for short gaps [28] and (2) the calculation of the secondary effect is extremely complicated, especially for the photoionization [29], which would bring about great difficulties of the PD sequence simulation. The detailed expressions of the above transport parameters

After the streamer arrives at the interface between the cavity and the dielectric, the charges will accumulate on the dielectric surface. We use the following equation to describe the transition

where ΔS and ΔV represent the area and volume of unit grid after meshing, respectively. Surface charge distribution is assumed to keep unchanged during the discharge process.

During the streamer development, the influence of space charges on the electric field should not be neglected, so Poisson's equation is employed to obtain the electric field within the cavity:

N<sup>p</sup> � N<sup>e</sup> � N<sup>n</sup>

σΔS ¼ N<sup>p</sup> � N<sup>e</sup> � N<sup>n</sup>

In general, the time step for FCT is chosen based on the electron drift velocity, however, which may not apply to the circumstance in our simulation model. It is because apart from the streamer development, its extinguishment process also needs to be obtained which is responsible for the accumulation of electrons and ions. However, the drift velocity of electrons is about 100 higher than that of ions, and the choice of time step must lead to the large increase of calculation consumption at the later stage of discharge when ion drift dominates. Instead, if it is chosen based on the ion drift velocity, the accuracy of the calculation cannot be guaranteed at the initial stage of discharge. Therefore, as a compromise, the time step is set according to whether there are any electrons within the cavity volume. In detail, during the initial stage of streamer development, it is determined by the electron drift velocity. After electrons completely accumulate at the interface, it depends on the drift velocity of a positive ion or a negative one (both are the same). The expression for the time step is

$$
\Delta t\_{\rm e,p} = 0.1 \frac{\Delta z}{\left| \mathbf{W}\_{\rm e,p} \right|^{\rm max}} \tag{15}
$$

where <sup>Δ</sup><sup>t</sup> is the time step, <sup>Δ</sup><sup>z</sup> the grid length along <sup>z</sup>-direction, and j j <sup>W</sup> max the maximum value of charge drift velocity within the cavity.

According to Pedersen's model, the apparent charges are determined by charge transportation within the cavity, which could be detected by pulse current method. However, due to the effect of dielectric barriers, the pulse obtained at the external circuit may not reflect the streamer propagation. So, we use Sato's equation to calculate the current due to free charge movement [34], as follows:

$$I = \frac{e}{\mathcal{U}\_\mathbf{a}} \iiint\_V \left( N\_\mathbf{P} \mathbf{W}\_\mathbf{p} - N\_\mathbf{e} \mathbf{W}\_\mathbf{e} - N\_\mathbf{n} \mathbf{W}\_\mathbf{e} \right) \cdot \mathbf{E}\_\mathbf{a} dV \tag{16}$$

at AC voltage has been studied by many authors [2, 4–7], and a comprehensive understanding about it has been obtained, the PD mechanism under DC voltage needs to be clarified. In this chapter, the DC voltage with an amplitude of 3200 V is applied to the anode, and the cathode is grounded all the time. Of course, this model is also applied to the circumstance of AC voltage

Numerical Modeling of Partial Discharge Development Process

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119

The process of PD development in the cavity consists of two stages: the streamer propagation and surface charge accumulation. Figures 4 and 5 show the temporal and spatial distribution of electrons and positive ions during this process, respectively. After discharge conditions are satisfied, the streamer is initiated near the lower surface of dielectric. With the help of applied field, electrons propagate toward the anode. At 0.72 ns, the head of streamer arrives at the upper surface of dielectric. Based on this, the streamer development velocity could be

Figure 4. Evolution of electron concentration distribution during the first PD (a) within the cavity volume (unit: cm<sup>3</sup>

).

and (b) on the upper surface of the cavity (unit: cm<sup>2</sup>

)

application.

3.2. Simulation results

3.2.1. A PD development process

where U<sup>a</sup> indicates the applied voltage, E<sup>a</sup> the applied field, and V the discharging volume.

On one hand, the field within the cavity should exceed a critical value so that a discharge may take place. Based on the ignition condition of streamer, the critical field is expressed as follows [35]:

$$E\_b = \frac{24410\left(\frac{p}{760}d\_\mathrm{g}\right) + 6730\sqrt{\frac{p}{760}d\_\mathrm{g}}}{d\_\mathrm{g}}\tag{17}$$

where P is in Torr. After a discharge takes place, electrons and ions accumulate on the dielectric surface. Due to the recombination of charges from gas, surface, and bulk conduction of dielectric, the accumulated surface charges will decay until the next discharge occurs. It is found from our previous experiments that the decaying discipline of surface charges could be expressed as [36].

$$\frac{\sigma\_{\rm P}}{\sigma\_{\rm P^0}} = \mathcal{e}^{\overline{\rm \prime}} \tag{18}$$

$$\frac{\sigma\_{\text{e}}}{\sigma\_{\text{e0}}} = \overline{\sigma^{\text{v}}} \tag{19}$$

where σp0 and σe0 indicate initial positive charge and electron density at dielectric surfaces, respectively. η<sup>p</sup> and η<sup>e</sup> equal 312.5 and 568.8 ms, both of which represent the surface charge decay time for positive ions and electrons. The negative ion is neglected because its concentration is much lower in comparison with electron and positive ions.

On the other hand, although free electrons from the volume ionization and surface emission are formulated, their supply shows a strong scholastic behavior. Hence, there is usually a time delay between the instant of application of an electric field in excess of the critical field and the onset of breakdown, which is called a discharge time lag (strictly speaking, it is a statistical time lag, but the formative time lag is very short for cavity discharge and could be neglected). In order to simplify the physical process of free electron production, the discharge time lag is introduced to our model. Some experimental and simulation results show that the discharge time lag is not completely random, but is subject to exponential distribution [37, 38], which is expressed as

$$\tau = \begin{cases} \ast & E\_\mathbf{z} < E\_\mathbf{b} \\ -\ln(1 - P\_\mathbf{d}) / \zeta & E\_\mathbf{z} \ge E\_\mathbf{b} \end{cases} \tag{20}$$

where P<sup>d</sup> indicates the discharge probability, which belongs to [0, 1) and is random, ζ the rate parameter of exponential distribution.

In terms of Eq. (17), the critical field for gas breakdown is calculated, and it equals 67,000 V/cm. In this case, the potential difference across the electrodes is 3130 V. Because the PD mechanism at AC voltage has been studied by many authors [2, 4–7], and a comprehensive understanding about it has been obtained, the PD mechanism under DC voltage needs to be clarified. In this chapter, the DC voltage with an amplitude of 3200 V is applied to the anode, and the cathode is grounded all the time. Of course, this model is also applied to the circumstance of AC voltage application.
