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AC Breakdown Voltage of Parallel Electrodes in Air
Introduction
This simulation model focuses on electrical breakdown in air at atmospheric pressure, where streamer breakdown is the dominant mechanism. Unlike glow discharges, streamer discharges are unstable and marked by a rapid rise in discharge current due to electron impact ionization (see Fig. 7.1 in Ref. 1).
The model computes the AC breakdown voltage between parallel electrodes in air using a charge transport formulation. While implemented in one dimension for simplicity, the approach can be generalized to other gases and extended to higher dimensions.
Model Definition
The Electric Discharge interface is used to compute the discharge current. 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.
Breakdown is detected as a sharp current rise triggered by electron avalanches. To capture this process, the model conducts a transient study with an AC voltage whose amplitude is gradually increased after several cycles. The simulation stops once the conductive discharge current surpasses a specified threshold.
Results and Discussion
Figure 1 shows the discharge current and applied voltage as a function of AC periods for three frequencies. The results indicate that the breakdown voltage are largely independent of frequency and remains close to the DC value, provided the frequency does not enter the RF range.
Figure 1: The discharge current as a function of AC periods for three frequencies.
Reference
1. Y.P. Raizer, Gas Discharges Physics, Springer, 1997.
Application Library path: Electric_Discharge_Module/Electrical_Breakdown/breakdown_voltage_ac
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  1D.
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 > Time Dependent with Initialization.
6
Click  Done.
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.
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
50[kV]
50000 V
Maximum voltage
d
1[cm]
0.01 m
Gap length
f
50[Hz]
50 Hz
Frequency
t
1/f/100
2E-4 s
Time
Vapp
V0*(t*f/10)*sin(2*pi*f*t)
3.1395 V
Applied voltage
Geometry 1
Interval 1 (i1)
1
In the Model Builder window, expand the Component 1 (comp1) > Geometry 1 node.
2
Right-click Geometry 1 and choose Interval.
3
In the Settings window for Interval, locate the Interval section.
4
In the table, enter the following settings:
Coordinates (cm)
0
d
5
Click  Build All Objects.
Add Material from Library
In the Home toolbar, click  Windows and choose Add Material from Library.
Add Material
1
Go to the Add Material window.
2
In the tree, select Electric Discharge > Gases > Air > Air [Morrow and Lowke, 1997].
3
Click the Add to Component button in the window toolbar.
Electric Discharge (edis)
Electrode 1
1
In the Physics toolbar, click  Attributes and choose Electrode.
2
Select Boundary 1 only.
Gas 1
In the Model Builder window, click Gas 1.
Electrode 2
1
In the Physics toolbar, click  Attributes and choose Electrode.
2
Select Boundary 2 only.
3
In the Settings window for Electrode, locate the Terminal section.
4
In the V0 text field, type Vapp.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Sequence Type section.
3
From the list, choose User-controlled mesh.
Distribution 1
1
In the Model Builder window, right-click Edge 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
From the Distribution type list, choose Predefined.
4
In the Number of elements text field, type 200.
5
In the Element ratio text field, type 5.
6
Select the Symmetric distribution checkbox.
7
Click  Build All.
Study 1
Step 2: Time Dependent
1
In the Model Builder window, under Study 1 click Step 2: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(1/f/100,1/f/20,10/f).
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
f (Frequency)
50 500 5000
Hz
Next, keep track of the conduction current. Note that the displacement current is not of interest for evaluating AC breakdown.
Definitions
Global Variable Probe 1 (var1)
1
In the Definitions toolbar, click  Probes and choose Global Variable Probe.
2
In the Settings window for Global Variable Probe, type Ic in the Variable name text field.
3
Locate the Expression section. In the Expression text field, type comp1.edis.gas1.ece2.Ic.
Study 1
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
Add a stop condition to terminate the simulation once a streamer breakdown takes place.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
In the Model Builder window, expand the Study 1 > Solver Configurations > Solution 1 (sol1) > Time-Dependent Solver 1 node.
4
Right-click Study 1 > Solver Configurations > Solution 1 (sol1) > Time-Dependent Solver 1 and choose Stop Condition.
5
In the Settings window for Stop Condition, locate the Stop Expressions section.
6
Click  Add.
7
In the table, enter the following settings:
Stop expression
Stop if
Active
Description
abs(comp1.Ic)>1e-5[A]
True (>=1)
√
Stop expression 1
8
Locate the Output at Stop section. From the Add solution list, choose Step after stop.
9
In the Model Builder window, click Study 1.
10
In the Settings window for Study, locate the Study Settings section.
11
Clear the Generate default plots checkbox.
12
In the Study toolbar, click  Compute.
Results
Discharge Current
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Discharge Current in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Parametric Solutions 1 (sol3).
4
Click to expand the Title section. From the Title type list, choose None.
5
Locate the Plot Settings section.
6
Select the x-axis label checkbox. In the associated text field, type Period.
7
Select the Two y-axes checkbox.
8
Select the y-axis label checkbox. In the associated text field, type Current, A.
9
Select the Secondary y-axis label checkbox. In the associated text field, type Voltage, kV.
Global 1
1
Right-click Discharge Current 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
Ic
A
Global Variable Probe 1
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type t*f.
6
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose Cycle.
7
Click to expand the Legends section. Find the Include subsection. Clear the Solution checkbox.
8
Clear the Description checkbox.
9
Find the Prefix and suffix subsection. In the Prefix text field, type f = eval(f,Hz,4) Hz.
10
In the Discharge Current toolbar, click  Plot.
Global 2
1
In the Model Builder window, right-click Discharge Current and choose Global.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (sol1).
4
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
Vapp
V
Applied voltage
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type t*f.
7
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
Vapp
kV
Applied voltage
8
Locate the y-Axis section. Select the Plot on secondary y-axis checkbox.
9
Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Cycle.
10
From the Positioning list, choose Interpolated.
Discharge Current
1
In the Model Builder window, click Discharge Current.
2
In the Settings window for 1D Plot Group, locate the Axis section.
3
Select the Manual axis limits checkbox.
4
In the x minimum text field, type 0.
5
In the x maximum text field, type 6.
6
In the y minimum text field, type -1e-6.
7
In the y maximum text field, type 1e-6.
8
In the Secondary y minimum text field, type -30.
9
In the Secondary y maximum text field, type 30.
10
Locate the Legend section. From the Position list, choose Upper left.
11
In the Discharge Current toolbar, click  Plot.