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MRI Birdcage Coil
Introduction
This example involves designing and optimizing a birdcage coil so it can provide a homogeneous magnetic field distribution for a Magnetic Resonance Imaging (MRI) system. This is usually done to MRI systems so they can generate higher resolution images.
The homogeneous magnetic field is obtained by quadrature excitation and an optimal value of lumped elements in the coil. To find the optimal value at the desired Larmor frequency with an air phantom, a parametric sweep is carried out for the capacitance of the coil’s lumped elements. The performance of the coil, loaded with a human head phantom, is also studied.
Figure 1: The birdcage coil is shielded by a cylindrical conductive shell. The front part of the shield is removed for visualization purposes. Absorbing boundaries are not included in this figure.
Model Definition
Figure 1 shows the geometry of the example, which consists of a bird cage coil placed around a human head phantom. Noticeably, there are a number of capacitors on the coil. These determine the resonant frequency of the coil and the homogeneity of the field it produces. The coil is placed inside an RF shield. The coil surfaces and the shield around the coil are assigned a Perfect Electric Conductor (PEC) condition. Lumped ports are used to provide quadrature excitation for the coil, while the coil's capacitors are defined using lumped elements.
The air domain around the coil is modeled using an air sphere. Scattering boundary conditions are used along the sphere's boundaries to prevent any reflections into the modeling domain from the outermost boundaries.
To obtain a homogeneous field at the Larmor frequency for an air phantom, the capacitance of the lumped elements in the coil are tuned using a parametric sweep. The circularity of the field is evaluated by estimating the axial ratio of the magnetic field around the phantom. The sum of the axial ratio in dB is evaluated by the line integration of the following quantity:
20log10((Bright+Bleft)/(Bright−Bleft))
Here, Bright and Bleft are the right- and left-hand rotating components of the magnetic field, respectively. The homogeneity of the field is quantified by evaluating the standard deviation of the electric field around the phantom.
The automatic mesh control option in the Electromagnetic Waves, Frequency Domain interface is used with the maximum mesh element size as 1/6 of free space wavelength. This example also estimates the homogeneity and circularity of the field in the coil when loaded with a human head phantom.
Results and Discussion
Figure 2 shows the magnetic field around the air phantom with an optimum value for the capacitance of the lumped elements at the Larmor frequency. The real part of the magnetic flux density is almost orthogonal to the imaginary part of it, which indicates the flux is rotating circularly.
Figure 2: Magnetic density norm distribution with an arrow plot of the real (red) and imaginary (blue) part of the magnetic flux density for the coil only.
Figure 3 shows the field for the coil loaded with the human head phantom. Compared to the case with an air phantom, the uniformity and circularity of the field is distorted, due to the high dielectric loading in the middle of the coil. The coil’s capacitors can be tuned further for this loaded case.
Figure 3: Magnetic density norm distribution with an arrow plot that shows the real (red) and imaginary (blue) part of the magnetic flux density when loaded with the human head phantom
Figure 4 and Figure 5 show the integration of the axial ratio of the magnetic flux density and the standard deviation of the electric field norm around the head model for different values of the lumped elements' capacitances. To achieve a homogeneous circularly-polarized magnetic field, it can be seen that the optimal value of the capacitance is around 28 pF.
Figure 4: The integration of the axial ratio of the magnetic flux density around the head model
Figure 5: The standard deviation of the electric field norm around the head model
Application Library path: RF_Module/Passive_Devices/mri_coil
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 Radio Frequency>Electromagnetic Waves, Frequency Domain (emw).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies>Frequency Domain.
6
Click  Done.
Study 1
Step 1: Frequency Domain
1
In the Model Builder window, under Study 1 click Step 1: Frequency Domain.
2
In the Settings window for Frequency Domain, locate the Study Settings section.
3
In the Frequencies text field, type 63.87[MHz].
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
c_value
10[pF]
1E-11 F
Capacitance used on the rungs
r_coil
0.24[m]
0.24 m
Radius of the coil
h_coil
0.3[m]
0.3 m
Height of the coil
l_element
0.01[m]
0.01 m
Length of the capacitive elements
Geometry 1
The model geometry is available as a parameterized geometry sequence in a separate MPH-file. If you want to build it from scratch, follow the instructions in the section Appendix: Geometry Modeling Instructions. Otherwise load it from file with the following steps.
1
In the Geometry toolbar, click  Insert Sequence.
2
Browse to the model’s Application Libraries folder and double-click the file mri_coil_geom_sequence.mph.
3
In the Geometry toolbar, click  Build All.
4
Click the  Zoom Extents button in the Graphics toolbar.
5
Click the  Wireframe Rendering button in the Graphics toolbar.
Import 1 (imp1)
1
In the Geometry 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 mri_coil.mphbin.
Form Union (fin)
1
In the Model Builder window, click Form Union (fin).
2
In the Settings window for Form Union/Assembly, locate the Form Union/Assembly section.
3
From the Repair tolerance list, choose Absolute.
4
In the Absolute repair tolerance text field, type 2.0E-9.
Adjusting the tolerance is required only if you build the geometry with CAD kernel. The default tolerance is fine if you use COMSOL kernel.
5
In the Geometry toolbar, click  Build All.
6
Click  Build All.
Create a set of selections to be used when setting up the physics. First, create a selection for the coil surfaces.
Definitions
Coil
1
In the Definitions toolbar, click  Explicit.
2
Click the  Zoom In button in the Graphics toolbar.
3
In the Settings window for Explicit, type Coil in the Label text field.
4
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
5
Click  Paste Selection.
6
In the Paste Selection dialog box, type 9 10 11 12 14 16 19 21 31 33 36 38 66 67 76 77 85 87 90 92 95 97 100 102 in the Selection text field.
7
Click OK.
Add a selection for the edges around the coil to evaluate the average field.
Circle
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Circle in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Edge.
4
Select Edge 61 only.
5
Select the Group by continuous tangent check box.
Add a selection for the absorbing boundaries surrounding the model domain.
Absorbing boundaries
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Absorbing boundaries in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundary 4 only.
5
Select the Group by continuous tangent check box.
Define the operators to evaluate the average field around the coil.
Average 1 (aveop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Average.
2
In the Settings window for Average, locate the Source Selection section.
3
From the Geometric entity level list, choose Edge.
4
From the Selection list, choose Circle.
Integration 1 (intop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, locate the Source Selection section.
3
From the Geometric entity level list, choose Edge.
4
From the Selection list, choose Circle.
Define the variables to evaluate the axial ratio of the magnetic field and the standard deviation of the electric field.
Variables 1
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Bleft
abs(emw.Bx+j*emw.By)
T
Left hand rotating component of magnetic flux
Bright
abs(emw.Bx-j*emw.By)
T
Right hand rotating component of magnetic flux
BaxialratiodB
20*log10((Bright+Bleft)/(Bright-Bleft))
 
Magnetic flux axial ratio
intBaxialratiodB
intop1(abs(BaxialratiodB))
m
Integration of magnetic flux circularity around the phantom
stdev
sqrt(aveop1(emw.normE^2)-aveop1(emw.normE)^2)
V/m
Standard deviation of E norm
View 1
Hide the outermost boundaries to view the interior parts when setting up the physics.
Hide for Physics 1
1
In the Model Builder window, right-click View 1 and choose Hide for Physics.
2
In the Settings window for Hide for Physics, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Click  Paste Selection.
5
In the Paste Selection dialog box, type 1-5, 8, 23, 26, 29, 63-65, 68-71 in the Selection text field.
6
Click OK.
Materials
Use the material properties of air for all the domains in the model.
Material 1 (mat1)
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
In the Settings window for Material, locate the Material Contents section.
3
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
Relative permeability
mur_iso ; murii = mur_iso, murij = 0
1
1
Basic
Electrical conductivity
sigma_iso ; sigmaii = sigma_iso, sigmaij = 0
0
S/m
Basic
Electromagnetic Waves, Frequency Domain (emw)
Set up the physics for the model. Define PEC condition for the coil boundaries and provide quadrature excitation for the coil using lumped ports. Make use of lumped elements to model the capacitors in the coil.
Perfect Electric Conductor 2
1
In the Model Builder window, under Component 1 (comp1) right-click Electromagnetic Waves, Frequency Domain (emw) and choose Perfect Electric Conductor.
2
In the Settings window for Perfect Electric Conductor, locate the Boundary Selection section.
3
From the Selection list, choose Coil.
Perfect Electric Conductor 3
1
In the Physics toolbar, click  Boundaries and choose Perfect Electric Conductor.
2
Select Boundaries 5, 6, 65, and 78 only.
Lumped Port 1
1
In the Physics toolbar, click  Boundaries and choose Lumped Port.
2
Click the  Zoom In button in the Graphics toolbar.
3
Select Boundary 37 only.
For the first port, wave excitation is on by default.
4
In the Settings window for Lumped Port, locate the Lumped Port Properties section.
5
In the V0 text field, type 200.
Lumped Port 2
1
In the Physics toolbar, click  Boundaries and choose Lumped Port.
2
Select Boundary 101 only.
3
In the Settings window for Lumped Port, locate the Lumped Port Properties section.
4
From the Wave excitation at this port list, choose On.
5
In the V0 text field, type 200.
6
In the θin text field, type pi/2.
Lumped Element 1
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 91 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 2
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 96 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 3
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 86 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 4
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 32 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 5
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 15 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 6
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 20 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 7
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 35 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 8
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 89 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 9
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 99 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 10
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 94 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 11
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 84 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 12
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 30 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 13
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 13 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 14
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 18 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 15
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 39 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 16
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 93 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 17
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 103 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 18
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 98 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 19
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 88 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 20
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 34 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 21
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 17 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Lumped Element 22
1
In the Physics toolbar, click  Boundaries and choose Lumped Element.
2
Select Boundary 22 only.
3
In the Settings window for Lumped Element, locate the Settings section.
4
From the Lumped element device list, choose Capacitor.
5
In the Celement text field, type c_value.
Scattering Boundary Condition 1
1
In the Physics toolbar, click  Boundaries and choose Scattering Boundary Condition.
2
In the Settings window for Scattering Boundary Condition, locate the Boundary Selection section.
3
From the Selection list, choose Absorbing boundaries.
Study 1
Add a parametric sweep for the capacitance of the lumped elements in the coil.
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
c_value (Capacitance used on the rungs)
 
pF
5
Click  Range.
6
In the Range dialog box, type 20 in the Start text field.
7
In the Step text field, type 0.5.
8
In the Stop text field, type 30.
9
Click Replace.
10
In the Study toolbar, click  Compute.
Results
Electric Field (emw)
Click the  Zoom Extents button in the Graphics toolbar.
Study 1/Solution 1 (sol1)
Add a selection for the domains around the phantom to visualize the fields.
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 3–6 only.
Add a slice plot for the magnetic field and arrow plots to view the direction of the magnetic field.
3D Plot Group 2
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 Parameter value (c_value (pF)) list, choose 28.5.
Slice 1
1
In the 3D Plot Group 2 toolbar, click  Slice.
2
In the Settings window for Slice, locate the Plane Data section.
3
From the Plane list, choose XY-planes.
4
In the Planes text field, type 1.
5
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Electromagnetic Waves, Frequency Domain>Magnetic>emw.normB - Magnetic flux density norm - T.
3D Plot Group 2
In the Model Builder window, click 3D Plot Group 2.
Arrow Volume 1
1
In the 3D Plot Group 2 toolbar, click  Arrow Volume.
2
In the Settings window for Arrow Volume, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Electromagnetic Waves, Frequency Domain>Magnetic>emw.Bx,...,emw.Bz - Magnetic flux density.
3D Plot Group 2
In the Model Builder window, click 3D Plot Group 2.
Arrow Volume 2
1
In the 3D Plot Group 2 toolbar, click  Arrow Volume.
2
In the Settings window for Arrow Volume, locate the Expression section.
3
In the X component text field, type imag(emw.Bx).
4
In the Y component text field, type imag(emw.By).
5
In the Z component text field, type imag(emw.Bz).
6
Locate the Coloring and Style section. From the Color list, choose Blue.
7
In the 3D Plot Group 2 toolbar, click  Plot.
8
Click the  Go to XY View button in the Graphics toolbar.
9
Click the  Zoom Extents button in the Graphics toolbar.
The plot shows the homogeneous and circularly polarized magnetic field around the air phantom. It is plotted in Figure 2.
1D Plot Group 3
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
Global 1
1
Right-click 1D Plot Group 3 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)>Definitions>Variables>intBaxialratiodB - Integration of magnetic flux circularity around the phantom - m.
1D Plot Group 3
1
In the Model Builder window, click 1D Plot Group 3.
2
In the 1D Plot Group 3 toolbar, click  Plot.
1D Plot Group 4
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
Global 1
1
Right-click 1D Plot Group 4 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)>Definitions>Variables>stdev - Standard deviation of E norm - V/m.
1D Plot Group 4
1
In the Model Builder window, click 1D Plot Group 4.
2
In the 1D Plot Group 4 toolbar, click  Plot.
It is obvious from the 1D plots that the optimum capacitance value for obtaining homogeneous magnetic field is around 28 pF. It is plotted in Figure 4 and Figure 5.
Global Definitions
Parameters 1
Now modify the capacitance of the lumped elements and rerun the model with the human head phantom.
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
c_value
28[pF]
2.8E-11 F
Capacitance used on the rungs
Materials
Material 2 (mat2)
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
Select Domains 5 and 6 only.
3
In the Settings window for Material, locate the Material Contents section.
4
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permittivity
epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0
40
1
Basic
Relative permeability
mur_iso ; murii = mur_iso, murij = 0
1
1
Basic
Electrical conductivity
sigma_iso ; sigmaii = sigma_iso, sigmaij = 0
0.9
S/m
Basic
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 Add Study in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Step 1: Frequency Domain
1
In the Settings window for Frequency Domain, locate the Study Settings section.
2
In the Frequencies text field, type 63.87[MHz].
3
In the Home toolbar, click  Compute.
Results
Study 2/Solution 2 (sol2)
In the Model Builder window, under Results>Datasets click Study 2/Solution 2 (sol2).
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
Add a selection for the domains around the phantom to visualize the fields.
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 3–6 only.
3D Plot Group 6
In the Results toolbar, click  3D Plot Group.
Slice 1
In the 3D Plot Group 6 toolbar, click  Slice.
3D Plot Group 6
1
In the Settings window for 3D Plot Group, locate the Data section.
2
From the Dataset list, choose Study 2/Solution 2 (sol2).
Slice 1
1
In the Model Builder window, click Slice 1.
2
In the Settings window for Slice, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Electromagnetic Waves, Frequency Domain>Magnetic>emw.normB - Magnetic flux density norm - T.
3
Locate the Plane Data section. From the Plane list, choose XY-planes.
4
In the Planes text field, type 1.
3D Plot Group 6
In the Model Builder window, click 3D Plot Group 6.
Arrow Volume 1
1
In the 3D Plot Group 6 toolbar, click  Arrow Volume.
2
In the Settings window for Arrow Volume, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Electromagnetic Waves, Frequency Domain>Magnetic>emw.Bx,...,emw.Bz - Magnetic flux density.
3D Plot Group 6
In the Model Builder window, click 3D Plot Group 6.
Arrow Volume 2
1
In the 3D Plot Group 6 toolbar, click  Arrow Volume.
2
In the Settings window for Arrow Volume, locate the Expression section.
3
In the X component text field, type imag(emw.Bx).
4
In the Y component text field, type imag(emw.By).
5
In the Z component text field, type imag(emw.Bz).
6
Locate the Coloring and Style section. From the Color list, choose Blue.
3D Plot Group 6
1
In the Model Builder window, click 3D Plot Group 6.
2
In the 3D Plot Group 6 toolbar, click  Plot.
The plot shows that the magnetic field is homogeneous and circularly polarized even when the coil is loaded with the human head phantom. It is plotted in Figure 3.
Global Evaluation 1
1
In the Results toolbar, click  Global Evaluation.
Evaluate the axial ratio of the magnetic field and the standard deviation of the electric field with the human head phantom.
2
In the Settings window for Global Evaluation, locate the Data section.
3
From the Dataset list, choose Study 2/Solution 2 (sol2).
4
Click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Component 1 (comp1)>Definitions>Variables>intBaxialratiodB - Integration of magnetic flux circularity around the phantom - m.
5
Click  Evaluate.
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
Click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Component 1 (comp1)>Definitions>Variables>stdev - Standard deviation of E norm - V/m.
5
Click  Evaluate.
Appendix: Geometry 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
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
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file mri_coil_parameters.txt.
5
In the table, enter the following settings:
Name
Expression
Value
Description
t_ring
0.015[m]
0.015 m
 
Geometry 1
Cylinder 1 (cyl1)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type r_coil.
4
In the Height text field, type h_coil.
5
Locate the Position section. In the z text field, type -h_coil/2.
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
t_ring
7
Clear the Layers on side check box.
8
Select the Layers on bottom check box.
9
Select the Layers on top check box.
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type -h_coil/2+t_ring.
4
Click  Show Work Plane.
Work Plane 1 (wp1)>Circle 1 (c1)
1
In the Work Plane toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type r_coil.
4
In the Sector angle text field, type 6.
5
Locate the Rotation Angle section. In the Rotation text field, type -22.5.
Work Plane 1 (wp1)>Convert to Curve 1 (ccur1)
1
In the Work Plane toolbar, click  Conversions and choose Convert to Curve.
2
Select the object c1 only.
3
In the Settings window for Convert to Curve, click  Build Selected.
Work Plane 1 (wp1)>Delete Entities 1 (del1)
1
In the Work Plane toolbar, click  Delete.
2
In the Settings window for Delete Entities, locate the Entities or Objects to Delete section.
3
From the Geometric entity level list, choose Boundary.
4
Click the  Zoom In button in the Graphics toolbar.
5
Click the  Zoom In button in the Graphics toolbar.
6
On the object ccur1, select Boundaries 2 and 3 only.
Extrude 1 (ext1)
1
In the Model Builder window, right-click Geometry 1 and choose Extrude.
2
In the Settings window for Extrude, locate the General section.
3
From the Input object handling list, choose Keep.
4
Locate the Distances section. In the table, enter the following settings:
Distances (m)
h_coil-2*t_ring
Extrude 2 (ext2)
1
In the Geometry toolbar, click  Extrude.
2
In the Settings window for Extrude, locate the Distances section.
3
In the table, enter the following settings:
Distances (m)
l_element
Move 1 (mov1)
1
In the Geometry toolbar, click  Transforms and choose Move.
2
Select the object ext2 only.
3
In the Settings window for Move, locate the Displacement section.
4
In the z text field, type 0 (h_coil-2*t_ring)/2-l_element/2 (h_coil-2*t_ring)-l_element.
Rotate 1 (rot1)
1
In the Geometry toolbar, click  Transforms and choose Rotate.
2
Select the objects ext1, mov1(1), mov1(2), and mov1(3) only.
3
In the Settings window for Rotate, locate the Rotation section.
4
In the Angle text field, type 0 45 90 135 180 225 270 315.
5
In the Geometry toolbar, click  Build All.
Convert to Surface 1 (csur1)
1
In the Geometry toolbar, click  Conversions and choose Convert to Surface.
2
Click in the Graphics window and then press Ctrl+A to select all objects.
3
In the Settings window for Convert to Surface, click  Build Selected.
Delete Entities 1 (del1)
1
In the Geometry toolbar, click  Delete.
2
In the Settings window for Delete Entities, locate the Entities or Objects to Delete section.
3
From the Geometric entity level list, choose Boundary.
4
Click the  Wireframe Rendering button in the Graphics toolbar.
5
On the object csur1, select Boundaries 3–6, 9, 10, 21, 22, 33, 34, 36, 39, 51, 52, 63, and 64 only.
Cylinder 2 (cyl2)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 0.15.
4
In the Height text field, type h_coil.
5
Locate the Position section. In the z text field, type -h_coil/2.
6
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
h_coil/2
7
Clear the Layers on side check box.
8
Select the Layers on top check box.
Cylinder 3 (cyl3)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type r_coil+0.1.
4
In the Height text field, type h_coil+0.1.
5
Locate the Position section. In the z text field, type -(h_coil+0.1)/2.
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 0.5.
4
In the Geometry toolbar, click  Build All.