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Self-Inductance and Mutual Inductance Between Single Conductors
Introduction
Two coils consisting of a single turn in a concentric coplanar arrangement are studied. Using a DC (steady state) analysis and an AC (frequency domain) analysis, the self-inductance of each coil and the mutual inductance between the two coils are computed using different approaches and are compared with analytical values. The two coils are excited in turns to compute all the elements of the inductance matrix.
Figure 1: Two concentric coplanar single-turn loops of wire. In each analysis, one of the two coils is excited, acting as the primary, while the other acts as the secondary.
Model Definition
The physical situation being modeled is shown in Figure 1. The two coils have radius R1 = 100 mm and R2 = 10 mm and are placed in a concentric and coplanar configuration. The wire radius is r0 = 1mm. The coils are here modeled in the 2D axisymmetric space, assuming no physical variation around the centerline. The coils are excited in turn with a prescribed current of 1 A.
In the limit as R1 >> R2 >> r0, the analytic expression for the mutual inductance between the two coils is:
where μ0 is the permeability of free space. This analytic expression is used to verify the accuracy of the model.
Figure 2: The concentric coils can be considered as a four-terminal device. The output can either be an open circuit, or a load can be applied.
Another way to consider this system is as a four-terminal device, as shown in Figure 2. A known current applied at the input terminals of the device, the primary coil, induces a voltage difference across the output terminals, the secondary coil. The objective of the AC model is to compute the voltage difference at the output for the open circuit case, and the induced currents for the closed circuit case.
The two concentric coils are modeled in a 2D axisymmetric sense, as shown schematically in Figure 3. The modeling domain is surrounded by a region of infinite elements, which is a way to truncate a domain which stretches to infinity. Although the thickness of the Infinite Element Domain is finite, it can be thought of as a domain of infinite extent.
The coils are both modeled using the Coil feature, which can be thought of as introducing an excitation across an infinitesimal slit in an otherwise continuous torus. Since each coil has a single turn and is made up of conductive material, the Single conductor model is used in the Coil feature. The feature can be used to excite the coil in all cases: the open circuit case, the closed torus case, as well as to model an external load. The primary coil is excited by specifying a current of 1 A.
Figure 3: A schematic representation of the 2D axisymmetric model of the concentric coils.
Although induced currents exist only if there is some variation in the driving current with respect to time, it is still possible to evaluate the inductance for this case from a DC analysis. Self-inductance (L11) and mutual inductance (L12) are defined as the total magnetic flux B passing through a surface whose edges define the primary and the secondary coil, respectively. That is:
Where I1 is the current passing through the primary coil, n is the vector normal to the surface, and the integral is taken over the surface defined by the coil in question. Since the B-field is computed from the magnetic vector potential:
It is possible to use Stokes’ theorem, which states that a surface integral of the curl of a field equals the line integral over the rim of the surface:
Where t is the unit tangent vector around the rim of the surface.
When solving the Magnetic Fields interface in a Stationary study step, these quantities are computed automatically for the coils present in the model. Cycling the feed over the coils (leaving zero current on the others) it is possible to extract the whole inductance matrix.
An alternative approach to the computation of the self-inductance in stationary situation is the energy method, which is based on the total magnetic energy in the system. With this approach, the self-inductance is defined as:
where Wm is the magnetic energy density, the current I1 is the current feeding the system and I2 is equal to zero. Correspondingly, L22 can be determined by feeding only the second coil.
Figure 4: The mutual inductance in the secondary coil can be evaluated by taking the surface integral of the magnetic flux through the coil, or the path integral of the magnetic vector potential.
For the AC case, a 1 kHz sinusoidally time-varying current is driving the primary coil. This can either induce currents in the secondary coil or induce a voltage difference if the coil is being modeled as an open circuit.
The secondary coil uses the Coil feature to model both the open circuit and the closed circuit case. To model the open circuit case, the current through the coil is specified to be 0 A. The Coil feature introduces a coil voltage that causes no current to flow.
On the other hand, to model the closed circuit case, the voltage drop across the coil is fixed at 0 V. Although this seems to imply a short circuit, the reactance of the copper coil is inherently included, so the case being modeled is analogous to a closed continuous loop of wire.
In the AC case, there is no analytic solution to compare against. At any nonzero frequency, capacitive effects start to appear, and the skin effect also starts to alter the effective resistance of the coils. The magnitude of these effects can only be evaluated with a frequency domain model. Although the DC case does provide good predictions of the behavior at low frequencies, it cannot completely predict behavior at higher frequencies. As additional physical objects such as cores are introduced, the need for a frequency domain model for accurate prediction becomes greater.
Results and Discussion
The DC magnetic flux is plotted in Figure 5. In the stationary case, the self-inductance and mutual inductance are computed automatically by integrating the magnetic vector potential as detailed above. The values are available as postprocessing variables, mf.LCoil_1 and mf.L_2_1, which can be evaluated in a Global Evaluation node. The computed self-inductance is compared with the one computed using the energy method, while the mutual inductance is verified against the analytical value, which is applicable in the limit R1 >> R2 >> r0.
Figure 5: Magnetic flux lines for the DC case.
A second stationary study is performed after switching the excitation in order to compute the self-inductance for the inner coil and the mutual inductance between the outer and the inner coil.
For the time harmonic case, the induced currents of the secondary coil (connected to an open circuit) is plotted in Figure 6. The average of the induced currents over the cross section is zero, that is, there is no net current flow through the coil.
Figure 6: The real part of the induced currents in the primary coil for the open circuit case. The net current flow over the cross section is zero, this is true independently of phase.
In the time harmonic (frequency domain) case, the mutual inductance is computed as:
(1)
The computed mutual inductance is 1.973 − 0.004i nH. The small imaginary component is due to the resistive effects. More explicitly, due to the finite conductivity, there are eddy current losses in the wires and the coil AC impedance (V/I), though mainly reactive, has a small resistive part.
The phase-dependent induced currents of the secondary coil for the closed circuit case is plotted in Figure 7. Note that in this case, the net current over the coil cross section is non-zero.
Figure 7: The real part of the induced currents in the secondary coil for the closed circuit case.
The total induced current around the secondary coil is − 0.01675 − 0.02677i, the imaginary component implies a reactive current. This value is verified by an estimate computed from DC values:
where L21 is the mutual inductance and Z2 = R2 + iωL2 is the impedance of the inner coil. Note that the phase term is excluded.
Application Library path: ACDC_Module/Tutorials,_Coils/mutual_inductance
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 Axisymmetric.
2
In the Select Physics tree, select AC/DC > Electromagnetic Fields > Magnetic Fields (mf).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Stationary.
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
r_wire
1[mm]
0.001 m
Radius, wire
R1
100[mm]
0.1 m
Radius, outer coil
R2
10[mm]
0.01 m
Radius, inner coil
M
(mu0_const*pi*R2^2)/(2*R1)
1.9739E-9 H
Analytic mutual inductance
I1
1[A]
1 A
Current, outer coil
I2
0[A]
0 A
Current, inner coil
Here, mu0_const a predefined COMSOL constant for the permeability in vacuum.
Geometry 1
Create a circle for the simulation domain. Define a layer in the circle where you will assign the Infinite Element Domain.
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 Sector angle text field, type 180.
4
In the Radius text field, type 1.75*R1.
5
Locate the Rotation Angle section. In the Rotation text field, type -90.
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
50[mm]
Create a circle for the outer coil.
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 r_wire.
4
Locate the Position section. In the r text field, type R1.
Then, create a circle for the inner coil.
Circle 3 (c3)
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 r_wire.
4
Locate the Position section. In the r text field, type R2.
Define the Infinite Element Domain to apply a coordinate transformation that mathematically stretches the layer to infinity. The Physics-Controlled Mesh creates a Swept mesh inside the Infinite Elementsdomains.
Definitions
Infinite Element Domain 1 (ie1)
1
In the Definitions toolbar, click  Infinite Element Domain.
2
Select Domains 1 and 3 only.
Magnetic Fields (mf)
Now, set up the physics. Assign a Coil feature on the outer and the inner coil. The outer coil will be initially fed with a current I1=1[A].
1
Click the  Zoom In button in the Graphics toolbar.
Domain Coil 1
1
In the Physics toolbar, click  Domains and choose Domain Coil.
2
Select Domain 5 only.
3
In the Settings window for Domain Coil, locate the Coil section.
4
In the Icoil text field, type I1.
Specify I2=0[A] current for the Coil feature assigned to the inner coil to model the open circuit case.
Domain Coil 2
1
In the Physics toolbar, click  Domains and choose Domain Coil.
2
Select Domain 4 only.
3
In the Settings window for Domain Coil, locate the Coil section.
4
In the Icoil text field, type I2.
Materials
Next, assign material properties. Use Air for all domains.
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 Built-in > Air.
4
Click the Add to Component button in the window toolbar.
Materials
Air (mat1)
Then, override the coil domains with copper.
Add Material
1
Go to the Add Material window.
2
In the tree, select AC/DC > Copper.
3
Click the Add to Component button in the window toolbar.
4
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
Copper (mat2)
Select Domains 4 and 5 only.
Mesh 1
1
Click the  Zoom Extents button in the Graphics toolbar.
2
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Build All.
Solve the first case where the outer coil (named 1) is fed and the inner (named 2) is open.
Study 1
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, locate the Study Settings section.
3
Clear the Generate default plots checkbox.
4
In the Study toolbar, click  Compute.
Results
In the Model Builder window, expand the Results node.
Study 1/Solution 1 (sol1)
Select only the domains not part of the Infinite Element Domain for better magnetic flux visualization.
1
In the Model Builder window, expand the Results > Datasets node, then click Study 1/Solution 1 (sol1).
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
Select Domains 2, 4, and 5 only.
Magnetic Flux Density DC
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Magnetic Flux Density DC in the Label text field.
Streamline 1
1
Right-click Magnetic Flux Density DC and choose Streamline.
2
In the Settings window for Streamline, locate the Streamline Positioning section.
3
From the Positioning list, choose Starting-point controlled.
4
From the Entry method list, choose Coordinates.
5
In the r text field, type range(0,0.9*R1/29,0.9*R1).
6
In the z text field, type 0.
7
Locate the Coloring and Style section. Find the Line style subsection. From the Type list, choose Tube.
Color Expression 1
1
Right-click Streamline 1 and choose Color Expression.
2
Click the  Zoom Extents button in the Graphics toolbar.
The resulting plot shows the magnetic flux lines for the DC case as in Figure 5.
Evaluate the self inductance of the external coil and the mutual inductance of the outer coil with respect to the inner. Some additional quantities are also computed to verify the results.
Global Evaluation 1
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Expressions section.
3
Use the Add Expression button or enter the information manually in order to obtain the following Expressions table:
Expression
Unit
Description
mf.LCoil_1
nH
External coil inductance
2*mf.intWm/1[A^2]
nH
Energy estimate for external coil inductance
mf.L_2_1
nH
Computed mutual inductance
M
nH
Analytical mutual inductance
4
Click  Evaluate.
Next, compute the self inductance of the inner coil and the mutual inductance of the inner coil with respect to the outer. Start by switching the currents in the coils.
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
I1
0[A]
0 A
Current, outer coil
I2
1[A]
1 A
Current, inner coil
Now add and solve a second study for this case. The solution previously computed will still be available in Study 1.
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 Studies subsection. In the Select Study tree, select General Studies > Stationary.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
1
In the Settings window for Study, locate the Study Settings section.
2
Clear the Generate default plots checkbox.
3
In the Study toolbar, click  Compute.
The quantities of interest are evaluated in the following steps.
Results
Global Evaluation 2
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Data section.
3
From the Dataset list, choose Study 2/Solution 2 (sol2).
4
Locate the Expressions section. Use the Add Expression button or enter the information manually in order to obtain the following Expressions table:
Expression
Unit
Description
mf.LCoil_2
nH
Internal coil inductance
2*mf.intWm/1[A^2]
nH
Energy estimate for internal coil inductance
mf.L_1_2
nH
Computed mutual inductance
M
nH
Analytical mutual inductance
5
Clicknext to  Evaluate, then choose New Table.
Table 2
1
Go to the Table 2 window.
Self and mutual inductance variables as computed above are derived via concatenated flux, which is defined as the line integral of the magnetic vector potential along the coil. This approach gives the best accuracy.
For simple geometries like the present one, concatenated flux can be also computed explicitly using its definition as the integral of magnetic flux through a surface, although this approach usually gives less accurate results.
Results
Cut Line 2D 1
1
In the Results toolbar, click  Cut Line 2D.
2
In the Settings window for Cut Line 2D, locate the Line Data section.
3
In row Point 2, set r to R2.
Cut Line 2D 2
1
In the Results toolbar, click  Cut Line 2D.
2
In the Settings window for Cut Line 2D, locate the Line Data section.
3
In row Point 2, set r to R1.
Line Integration 1
1
In the Results toolbar, click  More Derived Values and choose Integration > Line Integration.
2
In the Settings window for Line Integration, locate the Data section.
3
From the Dataset list, choose Cut Line 2D 1.
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
mf.Bz/I1
nH
 
5
Clicknext to  Evaluate, then choose New Table.
Line Integration 2
1
Right-click Line Integration 1 and choose Duplicate.
2
In the Settings window for Line Integration, locate the Data section.
3
From the Dataset list, choose Cut Line 2D 2.
4
Clicknext to  Evaluate, then choose Table 3 - Line Integration 1.
Cut Line 2D 1
1
In the Model Builder window, under Results > Datasets click Cut Line 2D 1.
2
In the Settings window for Cut Line 2D, locate the Data section.
3
From the Dataset list, choose Study 2/Solution 2 (sol2).
Cut Line 2D 2
1
In the Model Builder window, click Cut Line 2D 2.
2
In the Settings window for Cut Line 2D, locate the Data section.
3
From the Dataset list, choose Study 2/Solution 2 (sol2).
Line Integration 1
1
In the Model Builder window, under Results > Derived Values click Line Integration 1.
2
In the Settings window for Line Integration, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
mf.Bz/I2
nH
 
4
Clicknext to  Evaluate, then choose Table 3 - Line Integration 1.
Line Integration 2
1
In the Model Builder window, click Line Integration 2.
2
In the Settings window for Line Integration, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
mf.Bz/I2
nH
 
4
Clicknext to  Evaluate, then choose Table 3 - Line Integration 1.
Experimentally, mutual inductance is measured by feeding an AC signal in the primary coil and measuring the voltage induced in the open-circuit secondary coil. This procedure can be simulated by using a Frequency Domain study step. Start by setting the AC feed on Coil 1 and the open circuit (zero current) condition on Coil 2.
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
I1
1[A]
1 A
Current, outer coil
I2
0[A]
0 A
Current, inner coil
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 Studies subsection. In the Select Study tree, select General Studies > Frequency Domain.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 3
Step 1: Frequency Domain
1
In the Settings window for Frequency Domain, locate the Study Settings section.
2
In the Frequencies text field, type 1[kHz].
3
In the Model Builder window, click Study 3.
4
In the Settings window for Study, locate the Study Settings section.
5
Clear the Generate default plots checkbox.
6
In the Study toolbar, click  Compute.
Results
Study 3/Solution 3 (sol3)
Select the inner coil domain.
1
In the Model Builder window, under Results > Datasets click Study 3/Solution 3 (sol3).
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
Select Domain 4 only.
Current Open Circuit
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Current Open Circuit in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 3/Solution 3 (sol3).
4
Locate the Plot Settings section. From the View list, choose New view.
Surface 1
1
Right-click Current Open Circuit 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 > Current density - A/m² > mf.Jphi - Current density, phi-component.
3
In the Current Open Circuit toolbar, click  Plot.
4
Click the  Zoom Extents button in the Graphics toolbar.
Compare the reproduced plot with Figure 6.
Evaluate the mutual inductance using Equation 1.
Global Evaluation 3
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Data section.
3
From the Dataset list, choose Study 3/Solution 3 (sol3).
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
mf.VCoil_2/1[A]/mf.iomega
nH
 
5
Click  Evaluate.
Finally, simulate the system as it was a transformer with a short-circuited secondary winding. Specify a voltage of 0 V for the Coil feature assigned to the inner coil to model the short-circuit condition.
Magnetic Fields (mf)
Domain Coil 2
1
In the Model Builder window, under Component 1 (comp1) > Magnetic Fields (mf) click Domain Coil 2.
2
In the Settings window for Domain Coil, locate the Coil section.
3
From the Coil excitation list, choose Voltage.
4
In the Vcoil text field, type 0.
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 Studies subsection. In the Select Study tree, select General Studies > Frequency Domain.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 4
Step 1: Frequency Domain
1
In the Settings window for Frequency Domain, locate the Study Settings section.
2
In the Frequencies text field, type 1[kHz].
3
In the Model Builder window, click Study 4.
4
In the Settings window for Study, locate the Study Settings section.
5
Clear the Generate default plots checkbox.
6
In the Study toolbar, click  Compute.
Results
Study 4/Solution 4 (sol4)
Select the inner coil domain.
1
In the Model Builder window, under Results > Datasets click Study 4/Solution 4 (sol4).
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
Select Domain 4 only.
Current Closed Circuit
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Current Closed Circuit in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 4/Solution 4 (sol4).
4
Locate the Plot Settings section. From the View list, choose View 2D 2.
Surface 1
1
Right-click Current Closed Circuit 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 > Current density - A/m² > mf.Jphi - Current density, phi-component.
3
In the Current Closed Circuit toolbar, click  Plot.
4
Click the  Zoom Extents button in the Graphics toolbar.
The reproduced plot should look like Figure 7.
Evaluate the total induced current on the inner (secondary) coil. This quantity is related to static quantities, being in the simplest approximation iωM/(R2 + iωL2) times 1[A].
Global Evaluation 4
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Data section.
3
From the Dataset list, choose Study 4/Solution 4 (sol4).
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
mf.ICoil_2
A
Coil current
-mf.iomega*withsol('sol1',mf.L_2_1)/(withsol('sol2',mf.RCoil_2)+withsol('sol2',mf.LCoil_2)*mf.iomega)*mf.ICoil_1
A
 
The withsol operator is used to evaluate a quantity using a different study or solution step.
5
Click  Evaluate.