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Modeling of a 3D Inductor
Introduction
Inductors are used in many applications for low-pass filtering or for impedance matching of predominantly capacitive loads. They are used in a wide frequency range from near static up to several MHz. An inductor usually has a magnetic core to increase the inductance while keeping its size small. The magnetic core also reduces the electromagnetic interference with other devices as the magnetic flux tends to stay within it. Because there are only crude analytical or empirical formulas available to calculate impedances, computer simulations or measurements are necessary during the design stage. In general, inductor modeling is more complex than modeling resistors and capacitors but similar principles apply. The model geometry is designed using an external CAD software, then it is imported into the AC/DC Module for static and frequency domain analysis. The inductor geometry is shown in Figure 1.
Figure 1: The inductor geometry.
First a magnetostatic simulation is performed to get the DC inductance. At low frequencies capacitive effects are negligible. Thus, the relevant equivalent circuit model is an ideal inductor in a series with an ideal resistor. The inductance and the resistance are both computed in the magnetostatic simulation. At a high frequency, capacitive effects and skin effect become significant and the equivalent circuit model involves connecting an ideal capacitor in parallel with the DC circuit. The skin effect modifies the current distribution in the winding so the resistance increases and the inductance also changes. The circuit parameters are obtained by analyzing the frequency dependent impedance obtained from a frequency domain simulation. In this tutorial, the AC analysis is done up to the point when the frequency-dependent impedance is computed.
Model Definition
The application uses the Magnetic Fields interface, which supports stationary, transient, and frequency-domain modeling. The following table lists the material properties used in this application:
material Parameter
Copper winding
Core
Air
σ
5.998·107 S/m
0 S/m
0 S/m
εr
1
1
1
μr
1
103
1
The outer boundaries are set to the default magnetic insulation,
which from the inductive perspective is equivalent to a perfect electric conductor. In the magnetostatic analysis, the coil is modeled by a Coil feature, which computes the current flow by means of a Coil Geometry Analysis preprocessing step and then applies a total current of 1 A. For the frequency-domain analysis, a Lumped Port with a fixed a current of 1 A is applied to the feed gap instead.
Results and Discussion
The magnetostatic analysis yields an inductance of 0.11 mH and a DC resistance of 0.29 mΩ. Figure 2 shows the magnetic flux density norm and the direction of the current flow.
Figure 2: Magnetic flux density norm and current direction for the magnetostatic analysis.
In the static (DC) limit, the potential drop along the winding is purely resistive and could in principle be computed separately and before the magnetic flux density is computed. When increasing the frequency, inductive effects start to limit the current and skin effect makes it increasingly difficult to resolve the current distribution in the winding. At sufficiently high frequency, the current is mainly flowing in a thin layer near the conductor surface. When increasing the frequency further, capacitive effects come into play and current is flowing across the winding as displacement current density. When going through the resonance frequency, the device goes from behaving as an inductor to become predominantly capacitive. At the self resonance, the resistive losses peak due to the large internal currents. Figure 4 shows the surface current distribution at 1 MHz. Typical for high frequency, the currents are displaced toward the edges of the conductor.
Figure 3: Surface current density at 1 MHz (below the resonance frequency).
Figure 4 shows how the resistive part of the coil impedance peaks at the resonance frequency near 6 MHz whereas Figure 5 shows how the reactive part of the coil impedance changes sign and goes from inductive to capacitive when passing through the resonance.
Figure 4: The real part of the coil impedance peaks at the resonance frequency.
Figure 5: The reactive part of the coil impedance changes sign when passing through the resonance frequency, going from inductive to capacitive.
Application Library path: ACDC_Module/Inductive_Devices_and_Coils/inductor_3d
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  3D.
2
In the Select Physics tree, select AC/DC>Electromagnetic Fields>Magnetic Fields (mf).
3
Click Add.
4
In the Added physics interfaces tree, select Magnetic Fields (mf).
5
Click  Study.
6
In the Select Study tree, select General Studies>Stationary.
7
Click  Done.
Geometry 1
The main geometry is imported from file. Air domains are typically not part of a CAD geometry so they usually have to be added later. For convenience three additional domains have been defined in the CAD file. These are used to define a narrow feed gap where an excitation can be applied.
Import 1 (imp1)
1
In the Home toolbar, click  Import.
2
In the Settings window for Import, locate the Import section.
3
Click Browse.
4
Browse to the model’s Application Libraries folder and double-click the file inductor_3d.mphbin.
5
Click Import.
Sphere 1 (sph1)
1
In the Geometry toolbar, click  Sphere.
2
In the Settings window for Sphere, locate the Size section.
3
In the Radius text field, type 0.2.
4
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
0.05
5
Click  Build All Objects.
Form Union (fin)
1
In the Geometry toolbar, click  Build All.
2
Click the  Zoom Extents button in the Graphics toolbar.
3
Click the  Wireframe Rendering button in the Graphics toolbar.
The geometry should now look as in the figure below.
Materials
Next, define selections to be used when setting up materials and physics. Start by defining the domain group for the inductor winding and continue by adding other useful selections.
Definitions
Winding
1
In the Definitions toolbar, click  Explicit.
2
Select Domains 7, 8, and 14 only.
3
Right-click Explicit 1 and choose Rename.
4
In the Rename Explicit dialog box, type Winding in the New label text field.
5
Click OK.
Gap
1
In the Definitions toolbar, click  Explicit.
2
Select Domain 9 only.
3
Right-click Explicit 2 and choose Rename.
4
In the Rename Explicit dialog box, type Gap in the New label text field.
5
Click OK.
Core
1
In the Definitions toolbar, click  Explicit.
2
Select Domain 6 only.
3
Right-click Explicit 3 and choose Rename.
4
In the Rename Explicit dialog box, type Core in the New label text field.
5
Click OK.
Infinite Elements
1
In the Definitions toolbar, click  Explicit.
2
Select Domains 1–4 and 10–13 only.
3
Right-click Explicit 4 and choose Rename.
4
In the Rename Explicit dialog box, type Infinite Elements in the New label text field.
5
Click OK.
Nonconducting
1
In the Definitions toolbar, click  Explicit.
2
Select Domains 1–6 and 9–13 only.
3
Right-click Explicit 5 and choose Rename.
4
In the Rename Explicit dialog box, type Nonconducting in the New label text field.
5
Click OK.
Nonconducting without IE
1
In the Definitions toolbar, click  Explicit.
2
Select Domains 5, 6, and 9 only.
3
Right-click Explicit 6 and choose Rename.
4
In the Rename Explicit dialog box, type Nonconducting without IE in the New label text field.
5
Click OK.
Use infinite elements to emulate an infinite open space surrounding the inductor.
Infinite Element Domain 1 (ie1)
1
In the Definitions toolbar, click  Infinite Element Domain.
2
In the Settings window for Infinite Element Domain, locate the Domain Selection section.
3
From the Selection list, choose Infinite Elements.
4
Locate the Geometry section. From the Type list, choose Spherical.
Now define the material settings.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select AC/DC>Copper.
4
Click Add to Component in the window toolbar.
5
In the tree, select Built-in>Air.
6
Click Add to Component in the window toolbar.
7
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Copper (mat1)
1
In the Model Builder window, under Component 1 (comp1)>Materials click Copper (mat1).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Winding.
Air (mat2)
1
In the Model Builder window, click Air (mat2).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Nonconducting.
The core material is not part of the material library so it is entered as a user-defined material.
Core
1
In the Model Builder window, right-click Materials and choose Blank Material.
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Core.
4
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permeability
mur_iso ; murii = mur_iso, murij = 0
1e3
1
Basic
Electrical conductivity
sigma_iso ; sigmaii = sigma_iso, sigmaij = 0
0
S/m
Basic
Relative permittivity
epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0
1
1
Basic
5
Right-click Material 3 (mat3) and choose Rename.
6
In the Rename Material dialog box, type Core in the New label text field.
7
Click OK.
Magnetic Fields (mf)
Select Domains 1–8 and 10–14 only.
Coil 1
1
Right-click Component 1 (comp1)>Magnetic Fields (mf) and choose the domain setting Coil.
2
In the Settings window for Coil, locate the Domain Selection section.
3
From the Selection list, choose Winding.
4
In the Model Builder window, expand the Coil 1 node.
Input 1
1
In the Model Builder window, expand the Component 1 (comp1)>Magnetic Fields (mf)>Coil 1>Geometry Analysis 1 node, then click Input 1.
2
Select Boundary 58 only.
Geometry Analysis 1
In the Model Builder window, click Geometry Analysis 1.
Output 1
1
In the Physics toolbar, click  Attributes and choose Output.
2
Select Boundary 79 only.
The physics-controlled meshing functionality in the Magnetic Fields interface automatically creates a mesh suited for the requirements of the physics interface. In this case, the functionality creates a swept mesh in the infinite element region. This kind of mesh is able to handle the steep radial scaling with high accuracy using a limited number of elements.
Mesh 1
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Build All.
Add a Coil Geometry Analysis study step before the Stationary step to compute the direction of the current applied in the windings.
Study 1
Coil Geometry Analysis
1
In the Study toolbar, click  Study Steps and choose Other>Coil Geometry Analysis.
2
Right-click Study 1>Step 2: Coil Geometry Analysis and choose Move Up.
The magnetostatic model is now ready to solve.
3
In the Study toolbar, click  Compute.
Results
Magnetic Flux Density Norm (mf)
The default plot group shows the magnetic flux density norm and helps in detecting possible modeling errors.
Definitions
View 1
1
In the Model Builder window, under Component 1 (comp1)>Definitions click View 1.
2
In the Settings window for View, locate the View section.
3
Clear the Show grid check box.
Results
Additional plots, more specific to this problem, can be obtained by manipulating the datasets as described in the next paragraphs.
Study 1/Solution 1 (3) (sol1)
1
In the Model Builder window, expand the Results>Datasets node.
2
Right-click Results>Datasets>Study 1/Solution 1 (sol1) and choose Duplicate.
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Winding.
Study 1/Solution 1 (4) (sol1)
In the Model Builder window, under Results>Datasets right-click Study 1/Solution 1 (1) (sol1) and choose Duplicate.
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Core.
3D Plot Group 2
In the Results toolbar, click  3D Plot Group.
Streamline 1
1
Right-click 3D Plot Group 2 and choose Streamline.
2
In the Settings window for Streamline, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Magnetic Fields>Coil parameters>mf.coil1.eCoilx,...,mf.coil1.eCoilz - Coil direction.
3
Select Boundary 58 only.
Volume 1
1
In the Model Builder window, right-click 3D Plot Group 2 and choose Volume.
2
In the Settings window for Volume, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (4) (sol1).
4
In the 3D Plot Group 2 toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
6
Click the  Zoom In button in the Graphics toolbar.
Next, evaluate the coil inductance and resistance and compare them respectively to the inductance estimated from the total magnetic energy and to the resistance defined using Ohm’s law.
Global Evaluation 1
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
mf.RCoil_1
Ω
Coil resistance (DC)
mf.LCoil_1
H
Coil inductance
mf.VCoil_1/mf.ICoil_1
Ω
Voltage drop definition
2*mf.intWm/mf.ICoil_1^2
H
Inductance via magnetic energy density
4
Click  Evaluate.
Table
1
Go to the Table window.
The results should be about 0.11 mH for the inductance and 0.29 mΩ for the resistance.
Now, try solving the model without the infinite elements. Since most of the magnetic flux resides inside the core region, the effect of the Infinite Elements is rather limited. In absence of the core, the effect of using infinite elements would be more significative.
Magnetic Fields (mf)
1
In the Model Builder window, under Component 1 (comp1) click Magnetic Fields (mf).
2
Select Domains 5–8 and 14 only.
Study 1
In the Home toolbar, click  Compute.
Results
Global Evaluation 1
1
In the Model Builder window, under Results>Derived Values click Global Evaluation 1.
2
In the Settings window for Global Evaluation, click  Evaluate.
Component 1 (comp1)
Next, connect a simple circuit to the model.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select AC/DC>Electrical Circuit (cir).
4
Click Add to Component 1 in the window toolbar.
5
In the Home toolbar, click  Add Physics to close the Add Physics window.
Electrical Circuit (cir)
Change the coil excitation so it can connect to the circuit.
Magnetic Fields (mf)
Coil 1
1
In the Model Builder window, under Component 1 (comp1)>Magnetic Fields (mf) click Coil 1.
2
In the Settings window for Coil, locate the Coil section.
3
From the Coil excitation list, choose Circuit (current).
Electrical Circuit (cir)
In the Model Builder window, under Component 1 (comp1) click Electrical Circuit (cir).
Voltage Source 1 (V1)
1
In the Electrical Circuit toolbar, click  Voltage Source.
2
In the Settings window for Voltage Source, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
n
0
Resistor 1 (R1)
1
In the Electrical Circuit toolbar, click  Resistor.
2
In the Settings window for Resistor, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p
1
n
2
4
Locate the Device Parameters section. In the R text field, type 100[mohm].
In order to connect the circuit to the finite elements model use the dedicated circuit element.
External I vs. U 1 (IvsU1)
1
In the Electrical Circuit toolbar, click  External I vs. U.
2
In the Settings window for External I vs. U, locate the External Device section.
3
From the V list, choose Coil voltage (mf/coil1).
4
Locate the Node Connections section. In the table, enter the following settings:
Label
Node names
p
2
n
0
Study 1
In the Home toolbar, click  Compute.
Results
Next, evaluate the current.
Global Evaluation 2
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
mf.ICoil_1
A
Coil current
4
Click  Evaluate.
Table
1
Go to the Table window.
The current is limited to approximately 10 A by the external resistor, which is much larger than the internal resistance of the winding.
Results
3D Plot Group 2
Now set up the model for computing the frequency-dependent impedance.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Electrical Circuit (cir).
4
Find the Studies subsection. In the Select Study tree, select General Studies>Frequency Domain.
5
Click Add Study in the window toolbar.
6
In the Home toolbar, click  Add Study to close the Add Study window.
Definitions
At high frequency the skin depth of the conductor is so small that it is not convenient to resolve it. Instead of modeling the entire domain, use a lossy boundary condition with the appropriate material information, specified in a boundary Material node. Create also a Selection node to simplify the specification of the boundary condition.
Conductor Boundaries
1
In the Definitions toolbar, click  Explicit.
2
Select Domains 7, 8, and 14 only.
3
In the Settings window for Explicit, locate the Output Entities section.
4
From the Output entities list, choose Adjacent boundaries.
5
Right-click Explicit 7 and choose Rename.
6
In the Rename Explicit dialog box, type Conductor Boundaries in the New label text field.
7
Click OK.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select AC/DC>Copper.
4
Click Add to Component in the window toolbar.
5
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Copper 1 (mat4)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Geometric entity level list, choose Boundary.
3
From the Selection list, choose Conductor Boundaries.
Magnetic Fields (mf)
1
In the Model Builder window, under Component 1 (comp1) click Magnetic Fields (mf).
2
In the Settings window for Magnetic Fields, locate the Domain Selection section.
3
From the Selection list, choose Nonconducting without IE.
Impedance Boundary Condition 1
1
In the Physics toolbar, click  Boundaries and choose Impedance Boundary Condition.
2
In the Settings window for Impedance Boundary Condition, locate the Boundary Selection section.
3
From the Selection list, choose Conductor Boundaries.
Coil 1
The Coil feature no longer applies to an active domain and it has no effect. Disable it for clarity.
1
In the Model Builder window, right-click Coil 1 and choose Disable.
Without additional information about the external feeding, the electric potential is not a well defined quantity at high frequency. In order to provide missing information, use the Lumped Port boundary feature to excite the system.
Lumped Port 1
1
In the Physics toolbar, click  Boundaries and choose Lumped Port.
2
Select Boundaries 59–62 only.
Enter the geometrical parameters of the boundary.
3
In the Settings window for Lumped Port, locate the Lumped Port Properties section.
4
From the Type of lumped port list, choose User defined.
5
In the hport text field, type 0.024.
6
In the wport text field, type 0.046.
7
Specify the ah vector as
1
x
0
y
0
z
8
From the Terminal type list, choose Current.
Apart from the surface losses in the copper conductor, there will also be losses in the core due to eddy currents. These losses are introduced by specifying an effective complex permeability for the magnetic iron in the core, a quantity which is often available from the manufacturer. There is a specific constituive relation for this.
Ampère’s Law 2
1
In the Physics toolbar, click  Domains and choose Ampère’s Law.
2
In the Settings window for Ampère’s Law, locate the Domain Selection section.
3
From the Selection list, choose Core.
4
Locate the Constitutive Relation B-H section. From the Magnetization model list, choose Magnetic losses.
5
From the μ′ list, choose User defined. From the μ′′ list, choose User defined. In the μ′ text field, type 1200.
6
In the μ′′ text field, type 100.
Study 2
Step 1: Frequency Domain
Set up a frequency sweep from 1 MHz to 10 MHz in steps of 0.25 MHz.
1
In the Model Builder window, under Study 2 click Step 1: Frequency Domain.
2
In the Settings window for Frequency Domain, locate the Study Settings section.
3
From the Frequency unit list, choose MHz.
4
Click  Range.
5
In the Range dialog box, type 1 in the Start text field.
6
In the Stop text field, type 10.
7
In the Step text field, type 0.25.
8
Click Replace.
9
In the Home toolbar, click  Compute.
Results
Magnetic Flux Density Norm (mf) 1
Check for possible modeling errors by comparing the obtained results with the following figure.
Proceed to look at the surface current distribution in the winding.
Study 2/Solution 3 (6) (sol3)
In the Model Builder window, under Results>Datasets right-click Study 2/Solution 3 (sol3) and choose Duplicate.
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Conductor Boundaries.
3D Plot Group 4
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Dataset list, choose Study 2/Solution 3 (6) (sol3).
4
From the Parameter value (freq (MHz)) list, choose 1.
Surface 1
1
Right-click 3D Plot Group 4 and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Magnetic Fields>Currents and charge>mf.normJs - Surface current density norm - A/m.
3D Plot Group 4
1
In the Model Builder window, click 3D Plot Group 4.
2
In the 3D Plot Group 4 toolbar, click  Plot.
This is the surface current distribution.
Finish the modeling session by plotting the real and imaginary parts of the coil impedance.
1D Plot Group 5
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, locate the Data section.
3
From the Dataset list, choose Study 2/Solution 3 (5) (sol3).
Global 1
1
Right-click 1D Plot Group 5 and choose Global.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Magnetic Fields>Ports>mf.Zport_1 - Lumped port impedance - Ω.
3
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
real(mf.Zport_1)
ohm
Real part of impedance
4
In the 1D Plot Group 5 toolbar, click  Plot.
The resistive part of the coil impedance peaks at the resonance frequency.
5
Click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Magnetic Fields>Ports>mf.Zport_1 - Lumped port impedance - Ω.
6
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
real(mf.Zport_1)/10[kohm]
1
Real part of impedance [tens of kohm]
real(mf.Zport_1/mf.iomega)
mH
Imaginary part of impedance divided by the angular frequency [mH]
1D Plot Group 5
1
In the Model Builder window, click 1D Plot Group 5.
2
In the Settings window for 1D Plot Group, locate the Axis section.
3
Select the Manual axis limits check box.
4
In the x minimum text field, type 0.
5
In the x maximum text field, type 10.
6
In the y minimum text field, type -1.
7
In the y maximum text field, type 2.
8
In the 1D Plot Group 5 toolbar, click  Plot.
The reactive part of the coil impedance approaches the static inductance value at low frequency. With increasing frequency, the quantity increases and then changes sign at the resonance. This happens since the inductor is entering a capacitive regime where the net current is dominated by displacement currents flowing perpendiculary to the coil windings.
Before saving the application, specify a plot to be used as a thumbnail.