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Current-Voltage Characteristics of a Wire-to-Wire Corona Discharge
Introduction
Computing current-voltage (I-V) characteristics is crucial for understanding the behavior of corona discharges, particularly in applications involving high-voltage electrical systems. The I-V curve of a corona discharge often resembles that of a forward-biased diode, where the discharge current remains minimal until a certain threshold voltage is reached, beyond which the current increases significantly. In this specific case, the threshold is at 5 kV, beyond which a sharp rise in current is observed, indicative of the onset of a strong corona discharge.
This numerical model leverages a continuation solver to efficiently compute the I-V characteristics of a corona discharge generated by a wire-to-wire configuration. The continuation solver allows for stable and accurate tracing of the solution path across varying voltage levels, making it an effective tool for modeling such nonlinear electrical phenomena. The model is designed with flexibility in mind, offering a robust framework that can be easily adapted to study other discharge configurations and types, facilitating a wide range of applications in high-voltage engineering and plasma physics.
Model Definition
The Electric Discharge interface is used to simulate the corona discharge. The built-in charge transport model is used:
where
•
e, p, n denote electrons, positive ions, and negative ions
•
ni is the number density of the charge carrier (SI unit: 1/m3)
•
E is the electric field (SI unit: V/m)
•
zi denotes the carrier charge (SI unit: 1)
•
μi denotes the carrier mobility (SI unit: m2/(V·s))
•
wi is the drift velocity in the electric field (SI unit: m/s)
•
Di is the diffusion coefficient (SI unit: m2/s)
•
Ri is the reaction rate (SI unit: 1/(m3·s))
•
α is the ionization coefficient (SI unit: 1/m)
•
η is the attachment coefficient (SI unit: 1/m)
•
βep is the electron–ion recombination coefficient (SI unit: m3/s)
•
βpn is the ion–ion recombination coefficient (SI unit: m3/s)
The above transport equations are fully coupled with Poisson’s equation through the electric field and the space charge:
where e is the elementary charge.
To compute the lumped properties of the corona discharge, this model neglects photoionization within the domain and the effects of secondary electron emission at the cathode. Importantly, the model does not rely on any ad-hoc assumptions to simplify the ionization layer, ensuring it remains self-consistent and applicable to a wide range of discharge configurations.
Results and Discussion
Figure 1 shows current-voltage (I-V) characteristics. As can be seen, the discharge current increases significantly after 5 kV. Figure 2 plots the spatial distribution of space charge density under an applied voltage of 10 kV. As depicted, the majority of the space charge, predominantly composed of positive ions, is concentrated to the left of the first wire. This phenomenon occurs because the generation of positive charges between the two wires enhances the electric field near the left edge of the first wire, leading to increased charge generation in that region. This behavior results from the interplay between the Laplace field and the space charge effect. When the applied voltage is increased to 15 kV, the primary space charge distribution shifts to the region between the wires. Further increasing the voltage leads to convergence difficulties in the model, indicating the onset of streamer discharge breakdown. Under these conditions, the maximum electron density can surpass 108 cm-3, which serves as a basis for evaluating the breakdown voltage.
Figure 1: The current-voltage characteristics of the wire-to-wire corona discharge.
Figure 2: The distribution of space charge density when the application voltage is 10 kV.
Application Library path: Electric_Discharge_Module/Corona_Discharges/corona_discharge_iv_curve
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
2
In the Select Physics tree, select Electric Discharge > Electric Discharge (edis).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Physics Interfaces > Stationary with Initialization.
6
Click  Done.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
V0
10[kV]
10000 V
Applied voltage
d
10[cm]
0.1 m
Out-of-plane thickness
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose cm.
Circle 1 (c1)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type 0.01.
Circle 2 (c2)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type 0.1.
4
Click  Build All Objects.
5
Click the  Zoom Extents button in the Graphics toolbar.
6
Locate the Position section. In the x text field, type 1.
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 3.
4
In the Height text field, type 3.
5
Locate the Position section. From the Base list, choose Center.
6
In the x text field, type 0.5.
Rectangle 2 (r2)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 10.
4
In the Height text field, type 10.
5
Locate the Position section. From the Base list, choose Center.
6
In the x text field, type 0.5.
Difference 1 (dif1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the objects r1 and r2 only.
3
In the Settings window for Difference, locate the Difference section.
4
Click to select the  Activate Selection toggle button for Objects to subtract.
5
Select the objects c1 and c2 only.
6
Click  Build All Objects.
Electric Discharge (edis)
1
In the Model Builder window, under Component 1 (comp1) click Electric Discharge (edis).
2
In the Settings window for Electric Discharge, locate the Physical Model section.
3
In the dz text field, type d.
Gas 1
In the Model Builder window, under Component 1 (comp1) > Electric Discharge (edis) click Gas 1.
Electrode 1
1
In the Physics toolbar, click  Attributes and choose Electrode.
2
Select Boundaries 9–12 only.
3
In the Settings window for Electrode, locate the Terminal section.
4
In the V0 text field, type V0.
5
Locate the Charge Transport section. From the Boundary condition for positive ions list, choose Number density.
Gas 1
In the Model Builder window, click Gas 1.
Electrode 2
1
In the Physics toolbar, click  Attributes and choose Electrode.
2
Select Boundaries 13–16 only.
3
In the Settings window for Electrode, locate the Charge Transport section.
4
From the Boundary condition for electrons list, choose Number density.
5
From the Boundary condition for negative ions list, choose Number density.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Electric Discharge > Gases > Air > Air [Kang et al. 2003].
4
Right-click and choose Add to Component 1 (comp1).
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Mesh 1
1
In the Settings window for Mesh, locate the Sequence Type section.
2
From the list, choose User-controlled mesh.
Distribution 1
1
In the Model Builder window, right-click Free Triangular 1 and choose Distribution.
2
Click the  Select Box button in the Graphics toolbar.
3
Select Boundaries 9–12 only.
4
In the Settings window for Distribution, locate the Distribution section.
5
In the Number of elements text field, type 8.
Size 1
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 2 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element size checkbox. In the associated text field, type 0.04.
Size 2
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 1 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element size checkbox. In the associated text field, type 0.2.
Boundary Layers 1
In the Mesh toolbar, click  Boundary Layers.
Boundary Layer Properties
1
Click the  Select Box button in the Graphics toolbar.
2
In the Model Builder window, click Boundary Layer Properties.
3
Select Boundaries 9–16 only.
4
In the Settings window for Boundary Layer Properties, click  Build All.
Study 1
Step 1: Electrostatics Initialization
1
In the Model Builder window, under Study 1 click Step 1: Electrostatics Initialization.
2
In the Settings window for Electrostatics Initialization, click to expand the Study Extensions section.
3
Select the Auxiliary sweep checkbox.
4
Click  Add.
5
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
V0 (Applied voltage)
50
V
Step 2: Stationary
1
In the Model Builder window, click Step 2: Stationary.
2
In the Settings window for Stationary, click to expand the Study Extensions section.
3
Select the Auxiliary sweep checkbox.
4
Click  Add.
5
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
V0 (Applied voltage)
50 100 200 500 range(1000,1000,10000)
V
6
In the Study toolbar, click  Compute.
Results
Space Charge Density
1
In the Settings window for 2D Plot Group, locate the Plot Settings section.
2
Clear the Plot dataset edges checkbox.
3
In the Label text field, type Space Charge Density.
Surface 1
1
In the Model Builder window, expand the Space Charge Density node, then click Surface 1.
2
In the Settings window for Surface, locate the Coloring and Style section.
3
From the Color table list, choose Prism.
Space Charge Density
1
In the Model Builder window, click Space Charge Density.
2
In the Space Charge Density toolbar, click  Plot.
3
Click the  Zoom Extents button in the Graphics toolbar.
IV Curve
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type IV Curve in the Label text field.
Global 1
1
Right-click IV Curve and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
edis.I0_0/d
uA/m
Discharge Current
4
In the IV Curve toolbar, click  Plot.