Transient Modeling of a Capacitor in a Circuit

This example solves a transient model of a capacitor in combination with an external electrical circuit. The finite element model of the capacitor is combined with a circuit model of a voltage source and a resistor. A step change in voltage is applied, and the transient current through the capacitor is computed and compared to the analytic result.

Figure 1: A simple capacitor composed of a disk of dielectric with metal plates on either side and lead wires is connected to a circuit model of a voltage source and a resistor.

Model Definition

The modeled capacitor consists of two metal disks with leads separated by a disk of quartz glass with a relative permittivity εr = 4.2 and a small electric conductivity σ = 10-14 S/m (Figure 1). The model includes a region of surrounding air (εr = 1.0, σ = 5·10-15 S/m) to account for the fringing fields. The capacitor is connected to an external circuit composed of a voltage source and a resistor. Initially, the capacitor is in an equipotential state, with no potential difference between the plates. The voltage across the system is turned on instantaneously, and the potential fields and current through the device are computed.

Assume that the capacitor plates themselves are highly conductive, so that their total effective resistivity is much lower than that of the external resistor. Under this assumption, the electric potential in each of the plates is uniform at any instant in time.

A separate electrostatic analysis can be used to compute the capacitance of the device with the result C = 43.4 pF. The external resistor has a resistance of R = 1000 Ω. The analytic solution for the current through a resistor and capacitor in series is

where V0 is the applied voltage.

Since the quartz and the air have a low conductivity, displacement currents will be dominant in the beginning of the transient for about 1 μs. After that, conduction currents will become significant and the model will start to deviate from the analytic approximation. Because both displacement and conduction currents exist in this model, use the Electric Currents interface.

When solving a finite element model and a circuit model in combination, it is sometimes necessary to adjust the solver settings. Here, you solve the electric currents problem and the electric circuits problem using a coupled direct solver. This is the most robust solver combination, but also the most memory intensive one.

Results and Discussion

Figure 2 compares the model result for the current through the capacitor as a function of time for a unit change in the applied voltage with the analytic solution. As the figure shows, the agreement is very good.

The displacement and conduction current densities at a point midway between the capacitor plates are plotted in Figure 3. While the displacement current density drops off to zero, the induced conduction current density rises in time to a steady-state value. At first, the magnitude of the displacement current density is much higher than the magnitude of the conduction current density, which means that the device has only small leakage currents and losses at these time scales.

Finally, Figure 4 compares the model to the analytical approximation on a longer time scale. The analytical approximation can be improved by adding a lumped resistor, parallel to the capacitor, representing the nonzero conductivity of the insulators.

Figure 2: The current through the capacitor after a change in voltage is applied across the system.

Figure 3: The induced conduction and displacement current density in the quartz dielectric.

Figure 4: The current through the capacitor. After about 1 μs the model starts to deviate from the analytic solution.

ACDC_Module/Capacitive_Devices/capacitor_transient

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
R	1000[ohm]	1000 Ω	External resistor resistance
C	43.4[pF]	4.34E-11 F	Device capacitance
V0	1[V]	1 V	Applied voltage

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

First, create a cylinder for the model domain.

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 20.

Click

Choose wireframe rendering to get a better view of the interior parts.

Click the

button in the toolbar.

Then, add a cylinder for the disc of dielectric with the two metal plates.

Cylinder 2 (cyl2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 10.

In the text field, type 4.

Locate the section. In the text field, type 8.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (cm)
Layer 1	5[mm]

Clear the check box.

Select the check box.

Click

Finish the geometry by adding two cylinders for the leads.

Cylinder 3 (cyl3)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.75.

In the text field, type 8.

Cylinder 4 (cyl4)

Right-click and choose .

In the window for , locate the section.

In the text field, type 12.

Click

The result should look like the image above.

Electric Currents (ec)

The model is composed of a disc of dielectric material with metal plates on either side and two lead wires. To get a better view, hide some of the boundaries. Begin by selecting the interface, then add a node.

Definitions

Hide for Physics 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 1, 4, and 23 only.

Add a couple of terminals to the interface and connect them to the circuit and ground.

Electric Currents (ec)

Terminal 1

In the window, under right-click and choose the domain setting .

Select Domains 4 and 6 only.

In the window for , locate the section.

From the list, choose .

A ground boundary condition is applied to all surfaces surrounding the lower electrode. Simultaneously the domains are deselected from the physics. For such models, this is the suggested setup.

Definitions

Explicit 1

In the toolbar, click

Select Domains 2 and 5 only.

In the window for , locate the section.

From the list, choose .

Electric Currents (ec)

Ground 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the window, click .

Select Domains 1, 3, 4, and 6 only.

Create a lumped and a , and put them in series with the capacitor model.

Electrical Circuit (cir)

In the window, under click .

Resistor 1 (R1)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Label	Node names
p	1
n	0

Voltage Source 1 (V1)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Label	Node names
p	2
n	0

External I vs. U 1 (IvsU1)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Label	Node names
p	2
n	1

Locate the section. From the V list, choose .

Next, assign material properties to the model. Begin by specifying for all domains. Adjust its conductivity to 5e-15[S/m].

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

Materials

Air (mat1)

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	5e-15[S/m]	S/m	Basic

Override the dielectric disc with glass (quartz).

Add Material

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Glass (quartz) (mat2)

Select Domain 3 only.

Mesh 1

In the window, under right-click and choose .

Study 1

In the window, click .

In the window for , locate the section.

Clear the check box.

Step 1: Time Dependent

In the window, under click .

In the window for , locate the section.

Click

In the dialog box, type 2e-8 in the text field.

In the text field, type 2e-7.

Click .

For maximal robustness, before solving, apply a direct solver, and from error estimate. This is suggested for many time dependent problems. As exclude algebraic may be relaxing some of the decision of the time dependent stepping algorithm, a maximum time step is added, and maximum BDF order is decreased. This preserves accuracy of variables even when currents are nearly zero.

Solution 1 (sol1)

In the toolbar, click