Integrated Square-Shaped Spiral Inductor

This example presents a model of a microscale square inductor, used for LC bandpass filters in microelectromechanical systems (MEMS).

The purpose of the application is to calculate the self-inductance of the microinductor. Given the magnetic field, you can compute the self-inductance, L, from the relation

where Wm is the magnetic energy and I is the current. The application uses the Terminal boundary condition, which sets the current to 1 A and automatically computes the self-inductance. The self-inductance L becomes available as the L11 component of the inductance matrix.

Model Definition

The model geometry consists of the spiral-shaped inductor and the air surrounding it. Figure 1 below shows the inductor and air domains used in the model. The outer dimensions of the model geometry are around 0.3 mm.

Figure 1: Inductor geometry and the surrounding air.

The model equations are the following:

In the equations above, σ denotes the electrical conductivity, A the magnetic vector potential, V the electric scalar potential, Je the externally generated current density vector, μ0 the permittivity in vacuum, and μr the relative permeability.

The electrical conductivity in the coil is set to 106 S/m and 1 S/m in air. The conductivity of air is arbitrarily set to a small value in order to avoid singularities in the model, but the error becomes small as long as the value of the conductivity is small compared with the other conductivities in the model.

The constitutive relation is specified with the expression

where H denotes the magnetic field.

The boundary conditions are of three different types corresponding to the three different boundary groups; see Figure 2 (a), (b), and (c).

Figure 2: Boundaries with the same type of boundary conditions.

The boundary condition for the boundary highlighted in Figure 2 (a) is a magnetic insulation boundary with a terminal boundary condition. For the boundaries in Figure 2 (b), both magnetic and electric insulation prevail. The condition for the last boundary, Figure 2 (c), is magnetic insulation set to a constant electric potential of 0 V (ground).

Results

Figure 3 shows the electric potential in the inductor and the magnetic flux lines. The color of the flow lines represents the magnitude of the magnetic flux. As expected this flux is largest in the middle of the inductor.

Figure 3: Electric potential in the device and magnetic flux lines around the device.

The computed self-inductance is 0.75 nH.

ACDC_Module/Inductive_Devices_and_Coils/spiral_inductor

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

Import 1 (imp1)

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file spiral_inductor.mphbin.

Click

Click the

button in the toolbar.

This geometry would be relatively straightforward to create from scratch; here it is imported for convenience.

Materials

Conductor

In the window, under right-click and choose .

Right-click and choose .

In the dialog box, type Conductor in the text field.

Click .

Select Domain 2 only.

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	1e6	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic

Air

In the window, right-click and choose .

Right-click and choose .

In the dialog box, type Air in the text field.

Click .

Select Domain 1 only.

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	1	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic

Setting the conductivity to zero in the air would lead to a numerically singular problem. You can avoid this problem by using a small nonzero value. As 1 S/m is much less than the electric conductivity in the inductor, the fields will only be marginally affected.

Magnetic and Electric Fields (mef)

Magnetic Insulation 1

In the window, under click .

Terminal 1

In the toolbar, click

and choose .

Select Boundary 5 only.

In the window for , locate the section.

In the I0 text field, type 1.

Magnetic Insulation 1

In the window, click .

Ground 1

In the toolbar, click

and choose .

Select Boundary 76 only.

This concludes the boundary settings. Note that the boundaries that you have not assigned are electrically and magnetically insulated by default.

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Study 1

Click

Results

Magnetic Flux Density Norm (mef)

The default plot shows the electric potential distribution in three cross-sections. There are plenty of other ways of visualizing the solution. The following instructions detail how to combine an electric potential distribution plot on the surface of the inductor with a streamline plot of the magnetic flux density in the air surrounding it.

3D Plot Group 3

In the toolbar, click

and choose .

Surface 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Study 1/Solution 1 (sol1)

In the window, expand the node, then click .

Selection

In the toolbar, click

and choose .

Selecting the boundaries of the inductor is most effectively done by first selecting all boundaries, then removing the exterior boundaries of the air box from the selection.

In the window for , locate the section.

From the list, choose .

Select Boundaries 5–9, 11–74, and 76 only.

Streamline 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 3.

Locate the section. Find the subsection. From the list, choose .

In the text field, type 1e-6.

Select the check box.

Click to expand the section. From the list, choose .

In the toolbar, click

The condition on the exterior boundaries causes the field lines to bend and follow the contours of the box. This inevitably introduces a systematic error to the inductance computation. It would be possible to reduce this error by increasing the size of the box, or introducing an infinite element domain. Nevertheless, since the field is comparatively small near the surface of the box, the result is reasonably accurate already. Try visualizing the local magnitude of the field by having it decide the color of the streamlines.

Color Expression 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Streamline 1

In the window, click .

In the toolbar, click

Global Evaluation 1

In the toolbar, click

In the window for , click in the upper-right corner of the section. From the menu, choose .

Click

Table

Go to the window.

The inductance evaluates to 0.75 nH.