EM.Ferma Tutorial Lesson 8: Simulating Coaxial Cables - Revision history

Asabet at 15:58, 11 November 2015

2015-11-11T15:58:04Z

Asabet at 15:31, 11 November 2015

2015-11-11T15:31:35Z

Asabet at 16:59, 26 October 2015

2015-10-26T16:59:12Z

Kazem Sabet at 13:39, 14 September 2015

2015-09-14T13:39:37Z

Kazem Sabet: /* Analyzing a Coaxial Line with Dielectric Filling */

2015-09-14T13:31:31Z

‎Analyzing a Coaxial Line with Dielectric Filling

Kazem Sabet: /* Getting Started */

2015-09-13T21:50:43Z

‎Getting Started

Kazem Sabet at 21:50, 13 September 2015

2015-09-13T21:50:24Z

Kazem Sabet: /* Getting Started */

2015-09-13T21:49:39Z

‎Getting Started

Kazem Sabet at 21:45, 13 September 2015

2015-09-13T21:45:50Z

Kazem Sabet: Created page with "{{projectinfo|Tutorial| Simulating Coaxial Cables|STAT146.png|In this project, you will analyze coaxial transmission line structures using the quasi-static simulation engine.|..."

2015-09-13T21:44:03Z

Created page with "{{projectinfo|Tutorial| Simulating Coaxial Cables|STAT146.png|In this project, you will analyze coaxial transmission line structures using the quasi-static simulation engine.|..."

New page

{{projectinfo|Tutorial| Simulating Coaxial Cables|STAT146.png|In this project, you will analyze coaxial transmission line structures using the quasi-static simulation engine.|

*PEC Objects
*Dielectric Objects
*2D Solution Plane
*Quasi-Static Analysis
|All versions|{{download|http://www.emagtech.com|EM.Ferma Lesson 8|[[EM.Cube]] 14.10}} }}

===Objective:===

To characterize 2D coaxial [[Transmission Lines|transmission lines]] with and without dielectric cores using quasi-static analysis.

===What You Will Learn:===

In this tutorial lesson, you will build coaxial cable structures using concentric cylinders and will examine their field distributions. You will also become familiar with the concept of material hierarchy in [[EM.Cube]].

==Getting Started==

Open the [[EM.Cube]] application and switch to Static Module. Start a new project with the following attributes:

#Name: [[STATICLesson6]]
#Length Units: mm
#Frequency Units: N/A
#Center Frequency: N/A
#Bandwidth: N/A

A coaxial transmission line consists of two concentric metallic cylinders. The interior cylinder is typically solid and is called the inner conductor. The exterior cylinder is hollow and is called the outer conductor. The space between the two concentric cylinder is filled with a dielectric material. The dielectric filling can be air or any other material.

==Creating the Concentric Metallic Cylinders==

Create two PEC groups called PEC_1 and PEC_2 in the Navigation Tree and draw two cylinder objects with the following properties:

{| class="wikitable"
|-
! Object !! Group !! LCS Coordinates !! Radius !! Height !! Cap Ends?
|-
| Cylinder_1 || PEC_1 || (0, 0, 0) || 1mm || 10mm || Yes
|-
| Cylinder_2 || PEC_2 || (0, 0, 0) || 2.3mm || 10mm || No
|-
|}

Define a voltage source called VS_1 with a default voltage of 1V and associate it with the PEC_1 group. Your inner conductor is held at a voltage of 1V, while your outer conductor is held at zero potential. Also, define a horizontal (Z-directed) field sensor and place it at (0, 0, 5mm).

Next, right-click on the <b>2D Solution Planes</b> item in the Navigation Tree and select <b>2D Domain Settings...</b> from the contextual menu. In the 2D Static Domain dialog, check the checkbox labeled <b>Treat Structure As Longitudinally Infinite...</b>. Click the <b>Add</b> button of the dialog to open up the Add/Edit Solution Plane dialog. From the "associated Field Sensor" drop-down list, choose the only option Sensor_1. Check the box labeled <b>Perform 2D Quasi-Static Simulation</b>. Then, from the drop-down list labeled "Signal Source", select your voltage source VS_1. This procedure is identical to what you followed in Tutorial Lesson 4 for the microstrip line.

Open the Domain Settings Dialog, select the "Default" option for the Domain Type and set the coordinates of the minimum as well as the maximum corners of the domain box to (2mm, 2mm, 10mm). Also open the Mesh Settings dialog and set the number of cells to 200, 200 and 100, along the X, Y and Z directions, respectively. Note that the mesh density along the Z-axis does not matter because you will perform a 2D analysis of the cross section of your structure in the sensor plane only.

<table>
<tr>
<td>
[[Image:STAT130.png|thumb|400px|The geometry of the air-filled coaxial cable.]]
</td>
<td>
[[Image:STAT131.png|thumb|400px|The static mesh of the air-filled coaxial cable.]]
</td>
</tr>
</table>

==Running a Quasi-Static Simulation and Visualizing the Results==

Run a quasi-static analysis of your air-filled coaxial line. At the end of the simulation, the output message window reports the computed values of the characteristic impedance and effective permittivity of the transmission line:

Z0: 48.2803 Ohms

Epsilon_Effective: 1

The effective permittivity of 1 is expected as you have an air-filled two-conductor TEM transmission line. You can also view the electric field and electric potential results.

<table>
<tr>
<td>
[[Image:STAT132.png|thumb|390px|The electric field distribution plot of the air-filled coaxial line.]]
</td>
<td>
[[Image:STAT132A.png|thumb|320px|The vector plot of the total electric field distribution in the air-filled coaxial line.]]
</td>
<td>
[[Image:STAT133.png|thumb|390px|The electric potential distribution plot of the air-filled coaxial line.]]
</td>
</tr>
</table>

Open the Data Manager and plot the data files "Sensor_1_X_ETotal.DAT" and "Sensor_1_Y_ETotal.DAT" in EM.Grid as shown in the figures below. Given the circular symmetry of your coaxial structure, the electric field distributions along the X and Y directions are identical. Also note that the electric field lines are radially oriented pointing from the inner conductor towards the outer conductor.

<table>
<tr>
<td>
[[Image:STAT134.png|thumb|420px|The graph of the total electric field of the air-filled coaxial line along the X-axis.]]
</td>
<td>
[[Image:STAT135.png|thumb|420px|The graph of the total electric field of the air-filled coaxial line along the Y-axis.]]
</td>
</tr>
</table>

To validate your simulation results, let's take a look at the analytical formulas for a coaxial cable:

<math> Z_0 = \frac{60}{\sqrt{\epsilon_{r}}} ln(b/a) = \frac{60}{\sqrt{1.0}} ln(2.3) \approx 50\Omega </math>

<math> \Phi(r) = V_0 \left[ 1 - \frac{ln(r/a)}{ln(b/a)} \right] </math>

<math> E_r(r) = \frac{1}{r} \frac{V_0}{ln(b/a)} </math>

where a and b are the radius of the inner and outer conductors, respectively, and ε<sub>r</sub> is the relative permittivity of the dielectric filling. Substituting a = 1mm and b = 2.3mm, we obtain the following values for the electric field and potential, which agree quite well with [[EM.Cube]]'s computed values:

<math> \Phi(r = 1mm) = V_0 = 1V, \quad \quad \Phi(r = 2.3mm) = 0V </math>

<math> E_r(r = 1mm) = \frac{1}{0.001} \frac{1}{ln(2.3)} = 1200V/m, \quad \quad E_r(r = 2.3mm) = \frac{1}{0.0023} \frac{1}{ln(2.3)} = 522V/m</math>

From the above figures, you can see the 1/r decay of the electric field along the radial direction from the inner conductor to the outer conductor.

==Analyzing a Coaxial Line with Dielectric Filling==

In the this part of the tutorial lesson, you will analyze a coaxial cable with a dielectric core made of "Mica" with ε<sub>r</sub> = 5.4. Keep the same inner conductor cylinder of the previous part. But increase the radius of the outer conductor cylinder to 6.93mm. Also, define a new dielectric material group in the Navigation Tree and select Mica as the material from [[EM.Cube]]'s Material List. Under Dielectric_1, draw a new Cylinder of radius 6.93mm with both ends capped. The following table summarizes the geometrical [[parameters]]:

{| class="wikitable"
|-
! Object !! Group !! LCS Coordinates !! Radius !! Height !! Cap Ends?
|-
| Cylinder_1 || PEC_1 || (0, 0, 0) || 1mm || 10mm || Yes
|-
| Cylinder_2 || PEC_2 || (0, 0, 0) || 6.93mm || 10mm || No
|-
| Cylinder_3 || Dielectric_1 || (0, 0, 0) || 6.93mm || 10mm || Yes
|-
|}

Keep the same voltage source, field sensor and mesh density as in the previous part. Note that you used a solid dielectric cylinder instead of a hollow dielectric cylinder to model the dielectric core of your coaxial cable. This is because in [[EM.Cube]]'s Static and FDTD Modules, a material hierarchy system is in effect, in which a PEC has precedence over a dielectric material. This means that if a PEC object and a dielectric object overlap each other over some region in the space, the common overlapped regions is assumed to be PEC rather than dielectric.

{{Note| In [[EM.Cube]]'s Static Module, overlapped regions between PEC and dielectric objects are always assumed to be PEC rather than dielectric. This hierarchical rule make construction of concentric structures much easier.}}

<table>
<tr>
<td>
[[Image:STAT136.png|thumb|400px|The geometry of the dielectric-filled coaxial cable.]]
</td>
<td>
[[Image:STAT137.png|thumb|400px|The static mesh of the dielectric-filled coaxial cable.]]
</td>
</tr>
</table>

Run a quasi-static analysis of your dielectric-filled coaxial line. At the end of the simulation, the output message window reports the computed values of the characteristic impedance and effective permittivity of the transmission line:

Z0: 49.1214 Ohms

Epsilon_Effective: 5.3994

An effective permittivity of ε<sub>eff</sub> = ε<sub>r</sub> = 5.4 is expected as you have an homogenous two-conductor TEM transmission line. Visualize the electric field and electric potential distributions and plot the 2D graphs of the total electric field along the X-axis.

<table>
<tr>
<td>
[[Image:STAT138.png|thumb|390px|The electric field distribution plot of the dielectric-filled coaxial line.]]
</td>
<td>
[[Image:STAT140.png|thumb|320px|The vector plot of the total electric field distribution in the dielectric-filled coaxial line.]]
</td>
<td>
[[Image:STAT139.png|thumb|390px|The electric potential distribution plot of the dielectric-filled coaxial line.]]
</td>
</tr>
</table>

<table>
<tr>
<td>
[[Image:STAT141.png|thumb|420px|The graph of the total electric field of the dielectric-filled coaxial line along the X-axis.]]
</td>
</tr>
</table>

You can validate your simulation results using the analytical formulas for a dielectric-filled coaxial cable:

<math> Z_0 = \frac{60}{\sqrt{\epsilon_{r}}} ln(b/a) = \frac{60}{\sqrt{5.4}} ln(6.93) \approx 50\Omega </math>

==Considering a Partial Dielectric Filling==

In the last part of this tutorial lesson, you will analyze a coaxial cable with a partial dielectric filling. Open the property dialog of Cylinder_3 from the previous part and reduce its radius to 4mm. The following table summarizes the geometrical [[parameters]]:

{| class="wikitable"
|-
! Object !! Group !! LCS Coordinates !! Radius !! Height !! Cap Ends?
|-
| Cylinder_1 || PEC_1 || (0, 0, 0) || 1mm || 10mm || Yes
|-
| Cylinder_2 || PEC_2 || (0, 0, 0) || 6.93mm || 10mm || No
|-
| Cylinder_3 || Dielectric_1 || (0, 0, 0) || 4mm || 10mm || Yes
|-
|}

Keep the same voltage source, field sensor and mesh density as in the previous part. The material hierarchy rules apply here, too. So you can simply use a solid dielectric cylinder to model your partial dielectric filling.

<table>
<tr>
<td>
[[Image:STAT145.png|thumb|400px|The geometry of the partially dielectric-filled coaxial cable.]]
</td>
<td>
[[Image:STAT146.png|thumb|400px|The static mesh of the partially dielectric-filled coaxial cable.]]
</td>
</tr>
</table>

Run a quasi-static analysis of your dielectric-filled coaxial line. At the end of the simulation, the output message window still reports the computed values of the characteristic impedance and effective permittivity of the transmission line as Z0 = 142.278Ω and ε<sub>eff</sub> = 0.643592. However, these results are wrong, particularly the effective permittivity which cannot be less than one. This is because your partially filled coaxial cable is now an inhomogeneous waveguide and no longer supports a dominant TEM mode. As we mentioned in Tutorial Lesson 4, the quasi-static analysis of [[EM.Cube]]'s Static Module calculated the characteristic impedance and effective permittivity of [[Transmission Lines|transmission lines]] based on the dominant TEM or TEM-like mode.

Visualize the electric field and electric potential distributions and plot the 2D graphs of the total electric field along the X-axis. From the figures below, you can see that the electric field is zero inside the inner conductor and outside the outer conductor, as you would expect. Between the two conductors, the electric field is still radial as in the homogeneous case of the previous part. However, the field inside the dielectric cylinder is much smaller than the field in the air region. This can be explained based on the boundary condition for the normal field component at the interface between the dielectric and air media at r = 4mm, which requires that E<sub>n</sub><sup>air</sup> = ε<sub>r</sub> E<sub>n</sub><sup>dielectric</sup>.

<table>
<tr>
<td>
[[Image:STAT147.png|thumb|390px|The electric field distribution plot of the partially dielectric-filled coaxial line.]]
</td>
<td>
[[Image:STAT149.png|thumb|320px|The vector plot of the total electric field distribution in the partially dielectric-filled coaxial line.]]
</td>
<td>
[[Image:STAT148.png|thumb|390px|The electric potential distribution plot of the partially dielectric-filled coaxial line.]]
</td>
</tr>
</table>

<table>
<tr>
<td>
[[Image:STAT150.png|thumb|420px|The graph of the total electric field of the partially dielectric-filled coaxial line along the X-axis.]]
</td>
</tr>
</table>

{{Note|The computed characteristic impedance and effective permittivity of [[Transmission Lines]] at the end of a 2D quasi-static analysis are valid only for homogenous multi-conductor structures that support a dominant TEM or TEM-like propagating mode.}}

{{EMCUBE directory}}

[[EM.Cube | Back to EM.Cube Wiki Main Page]]

@@ Line 5: / Line 5: @@
 *2D Solution Plane
 *Quasi-Static Analysis
-|All versions|{{download|http://www.emagtech.com/downloads/ProjectRepo/EMFerma_Lesson8.zip EM .Ferma Lesson 8}} }}
+|All versions|{{download|http://www.emagtech.com/downloads/ProjectRepo/EMFerma_Tutorial_6.zip EM .Ferma Lesson 8}} }}
 === What You Will Learn ===

@@ Line 5: / Line 5: @@
 *2D Solution Plane
 *Quasi-Static Analysis
-|All versions|{{download|http://www.emagtech.com/downloads/ProjectRepo/EMFerma_Tutorial_8.zip EM .Ferma Lesson 8}} }}
+|All versions|{{download|http://www.emagtech.com/downloads/ProjectRepo/EMFerma_Lesson8.zip EM .Ferma Lesson 8}} }}
 === What You Will Learn ===

@@ Line 7: / Line 7: @@
 |All versions|{{download|http://www.emagtech.com|EM.Ferma Lesson 8|[[EM.Cube]] 14.10}} }}
-===Objective:===
+=== What You Will Learn ===
-To characterize 2D coaxial [[Transmission Lines|transmission lines]] with and without dielectric cores using quasi-static analysis.
+In this tutorial you will build coaxial cable structures with and without dielectric cores using concentric cylinders and will examine their field distributions and transmission line characteristics. You will also become familiar with the concept of material hierarchy in [[EM.Ferma]].
-===What You Will Learn:===
+A coaxial transmission line consists of two concentric metallic cylinders. The interior cylinder is typically solid and is called the inner conductor. The exterior cylinder is hollow and is called the outer conductor. The space between the two concentric cylinder is filled with a dielectric material. The dielectric filling can be air or any other material.
-−
-In this tutorial lesson, you will build coaxial cable structures using concentric cylinders and will examine their field distributions. You will also become familiar with the concept of material hierarchy in [[EM.Cube]].
 ==Getting Started==
-Open the [[EM.Cube]] application and switch to Static Module. Start a new project with the following attributes:
+Open the [[EM.Cube]] application and switch to [[EM.Ferma]]. Start a new project with the following attributes:
-#Name: STATICLesson8
+<div class="noprint" style="float:left;margin-right:10px">
-#Length Units: mm
+{| border="1" class="wikitable"
-#Frequency Units: N/A
+|+ Starting [[Parameters]]
-#Center Frequency: N/A
+|-
-#Bandwidth: N/A
+! Name
-A coaxial transmission line consists of two concentric metallic cylinders. The interior cylinder is typically solid and is called the inner conductor. The exterior cylinder is hollow and is called the outer conductor. The space between the two concentric cylinder is filled with a dielectric material. The dielectric filling can be air or any other material.
+<div class="noprint" style="clear:both"></div>
 ==Creating the Concentric Metallic Cylinders==

@@ Line 176: / Line 176: @@
 <math> Z_0 = \frac{60}{\sqrt{\epsilon_{r}}}  ln(b/a) = \frac{60}{\sqrt{5.4}} ln(6.93) \approx 50\Omega </math>
 {{EMCUBE directory}}
 [[EM.Cube  | Back to EM.Cube Wiki Main Page]]

@@ Line 24: / Line 24: @@
 #Center Frequency: N/A
 #Bandwidth: N/A
 A coaxial transmission line consists of two concentric metallic cylinders. The interior cylinder is typically solid and is called the inner conductor. The exterior cylinder is hollow and is called the outer conductor. The space between the two concentric cylinder is filled with a dielectric material. The dielectric filling can be air or any other material.
@@ Line 31: / Line 30: @@
 Create two PEC groups called PEC_1 and PEC_2 in the Navigation Tree and draw two cylinder objects with the following properties:
 {| class="wikitable"
@@ Line 63: / Line 61: @@
 Run a quasi-static analysis of your air-filled coaxial line. At the end of the simulation, the output message window reports the computed values of the characteristic impedance and effective permittivity of the transmission line:
 Z0: 48.2803 Ohms
 Epsilon_Effective: 1
 The effective permittivity of 1 is expected as you have an air-filled two-conductor TEM transmission line. You can also view the electric field and electric potential results.
@@ Line 100: / Line 96: @@
 To validate your simulation results, let's take a look at the analytical formulas for a coaxial cable:
 <math> Z_0 = \frac{60}{\sqrt{\epsilon_{r}}}  ln(b/a) = \frac{60}{\sqrt{1.0}} ln(2.3) \approx 50\Omega </math>
 <math> \Phi(r) = V_0 \left[ 1 - \frac{ln(r/a)}{ln(b/a)} \right] </math>
 <math> E_r(r) =  \frac{1}{r} \frac{V_0}{ln(b/a)} </math>
 where a and b are the radius of the inner and outer conductors, respectively, and &epsilon;<sub>r</sub> is the relative permittivity of the dielectric filling. Substituting a = 1mm and b = 2.3mm, we obtain the following values for the electric field and potential, which agree quite well with [[EM.Cube]]'s computed values:
 <math> \Phi(r = 1mm) = V_0 = 1V, \quad \quad \Phi(r = 2.3mm) = 0V </math>
 <math> E_r(r = 1mm) = \frac{1}{0.001} \frac{1}{ln(2.3)}  = 1200V/m, \quad \quad E_r(r = 2.3mm) = \frac{1}{0.0023} \frac{1}{ln(2.3)}  = 522V/m</math>
 From the above figures, you can see the 1/r decay of the electric field along the radial direction from the inner conductor to the outer conductor.
 ==Analyzing a Coaxial Line with Dielectric Filling==
@@ Line 127: / Line 115: @@
 In the this part of the tutorial lesson, you will analyze a coaxial cable with a dielectric core made of "Mica" with &epsilon;<sub>r</sub> = 5.4. Keep the same inner conductor cylinder of the previous part. But increase the radius of the outer conductor cylinder to 6.93mm. Also, define a new dielectric material group in the Navigation Tree and select Mica as the material from [[EM.Cube]]'s Material List. Under Dielectric_1, draw a new Cylinder of radius 6.93mm with both ends capped. The following table summarizes the geometrical [[parameters]]:
 {| class="wikitable"
 |-

@@ Line 42: / Line 42: @@
 |-
 |}
 Define a voltage source called VS_1 with a default voltage of 1V and associate it with the PEC_1 group. Your inner conductor is held at a voltage of 1V, while your outer conductor is held at zero potential. Also, define a horizontal (Z-directed) field sensor and place it at (0, 0, 5mm).
@@ Line 60: / Line 59: @@
 </tr>
 </table>
 ==Running a Quasi-Static Simulation and Visualizing the Results==
@@ Line 73: / Line 71: @@
 The effective permittivity of 1 is expected as you have an air-filled two-conductor TEM transmission line. You can also view the electric field and electric potential results.
 <table>
@@ Line 142: / Line 139: @@
 |-
 |}
 Keep the same voltage source, field sensor and mesh density as in the previous part. Note that you used a solid dielectric cylinder instead of a hollow dielectric cylinder to model the dielectric core of your coaxial cable. This is because in [[EM.Cube]]'s Static and FDTD Modules, a material hierarchy system is in effect, in which a PEC has precedence over a dielectric material. This means that if a PEC object and a dielectric object overlap each other over some region in the space, the common overlapped regions is assumed to be PEC rather than dielectric.
 {{Note| In [[EM.Cube]]'s Static Module, overlapped regions between PEC and dielectric objects are always assumed to be PEC rather than dielectric. This hierarchical rule make construction of concentric structures much easier.}}
 <table>
@@ Line 161: / Line 156: @@
 Run a quasi-static analysis of your dielectric-filled coaxial line. At the end of the simulation, the output message window reports the computed values of the characteristic impedance and effective permittivity of the transmission line:
 Z0: 49.1214 Ohms
 Epsilon_Effective: 5.3994
 An effective permittivity of &epsilon;<sub>eff</sub> = &epsilon;<sub>r</sub> = 5.4 is expected as you have an homogenous two-conductor TEM transmission line. Visualize the electric field and electric potential distributions and plot the 2D graphs of the total electric field along the X-axis.
@@ Line 191: / Line 184: @@
 </tr>
 </table>
 You can validate your simulation results using the analytical formulas for a dielectric-filled coaxial cable:
 <math> Z_0 = \frac{60}{\sqrt{\epsilon_{r}}}  ln(b/a) = \frac{60}{\sqrt{5.4}} ln(6.93) \approx 50\Omega </math>
 {{EMCUBE directory}}
 [[EM.Cube  | Back to EM.Cube Wiki Main Page]]