View source for EM.Tempo Tutorial Lesson 7: Designing A Pyramidal Horn Antenna

{{projectinfo|Tutorial|Designing A Pyramidal Horn Antenna|Tempo L7 Fig title.png|In this project, pyramidal horn antennas are constructed and analyzed in EM.Tempo.|
*Waveguide Port Source
*Field Sensors
*Radiation Pattern
*Mesh Size
*Python Script
|All versions|{{download|http://www.emagtech.com/downloads/ProjectRepo/EMTempo_Lesson7.zip EMTempo_Lesson7}} }}

== What You Will Learn ==

In this tutorial you will build pyramidal horn antennas with different geometrical properties and aspect ratios and will analyze their radiation properties. You will also learn to use [[EM.Cube]]'s Python command line and will write you own Python scripts to perform some mathematical calculations.

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[[Image:Back_icon.png|30px]] '''[[EM.Cube#EM.Tempo_Documentation | Back to EM.Tempo Tutorial Gateway]]'''

[[Image:Download2x.png|30px]] '''[http://www.emagtech.com/downloads/ProjectRepo/EMTempo_Lesson7.zip Download projects related to this tutorial lesson]'''

== Getting Started ==

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

<div class="noprint" style="float:left;margin-right:10px">
{| border="1" class="wikitable"
|+ Starting Parameters
|-
! Name
| EMTempo_Lesson7
|-
! Length Units
| Centimeters
|-
! Frequency Units
| GHz
|-
! Center Frequency
| 1.2GHz
|-
! Bandwidth
| 0.6GHz
|}
</div>

<div class="noprint" style="clear:both"></div>

== Building the Pyramidal Horn Geometry ==

Click on the <b>Horn Antenna Wizard</b> [[Image:HornAntennaWizardIcon.png]] button of the Wizard Toolbar or select the menu item '''Tools &rarr; Antenna Wizards &rarr; Pyramidal Horn Antenna'''.

<table>
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<td> [[Image:Tempo L7 Fig1.png|thumb|720px|EM.Tempo's Object Toolbar.]] </td>
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</table>

The geometry of a pyramidal horn antenna with a rectangular waveguide feed appears in the project workspace. You are going to change the waveguide dimensions to those of a standard WR-650 waveguide. Open the variables dialog and make the following changes:

{| class="wikitable" style="text-align: center;"
|-
! scope="col"| Variable Name
! scope="col"| Original Definition
! scope="col"| New Definition
|-
| wg_a
| waveguide_design(1.0,fc)*1.1*to_meters 
| 16.51  
|-
| wg_l
| 0.5*lambda0_unit 
| 20
|-
| wg_off
| 0.25*lambda0_unit 
| 10
|}

<table>
<tr>
<td> [[Image:Tempo L7 Fig2.png|thumb|480px|The variables dialog showing the new waveguide feed dimensions.]] </td>
</tr>
</table>

<table>
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<td> [[Image:Tempo L7 Fig3.png|thumb|550px|The geometry of the pyramidal horn antenna after changing the waveguide feed dimensions.]] </td>
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</table>

The wizard already defined a waveguide port source in the middle of the waveguide feed as well as two orthogonal near-field sensor planes and a far-field radiation pattern observable.

== Running an FDTD Simulation of the Horn Antenna ==

Before running an FDTD analysis of your structure, let's take a close look at its mesh. Generate and view the mesh of the horn antenna with the default mesh density of 20 cells per wavelength. Since your project's center frequency is 1.2GHz with a bandwidth of 0.6GHz, the highest frequency of the project is 1.5GHz with a free-space wavelength of 20cm. Once the mesh appears on the screen, pay attention to [[EM.Tempo]]’s Status Bar. Next to the units and frequency sections of the status bar, you will see a number triplet: 183 × 150 × 113, in this case. These are the numbers of FDTD mesh grid lines inside the computational domain along the X, Y and Z directions, respectively. If you subtract each number by one, you will get the number of mesh cells in each direction. It is important to keep in mind that besides the visible computational domain in the project workspace, the FDTD simulation engine also sees a number of PML layers behind the domain walls. By default, there are six 8-layer CMPL walls embracing your computational domain. Therefore, your FDTD project involves a total of (183 -1 + 2×8) × (150 -1 + 2×8) × (113 -1 + 2×8) = 197 × 164 × 127 = 4,181,760 mesh cells. 

<table>
<tr>
<td> [[Image:Tempo L7 Fig10.png|thumb|550px|The FDTD mesh of the pyramidal horn antenna.]] </td>
</tr>
</table>

{{Note| When simulating very large computational problems, [[EM.Tempo]]'s GPU-accelerated FDTD engine can speed up the solution significantly if your computer has a powerful GPU card with enough onboard RAM memory.}} 

Run an FDTD Analysis of your horn antenna structure and examine the electric and magnetic field distributions on the two sensor planes. 

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<td>
[[Image:Tempo L7 Fig4.png|thumb|left|550px|Electric field distribution on FSX plane.]]
</td>
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<td>
[[Image:Tempo L7 Fig5.png|thumb|left|550px|Magnetic field distribution on FSX plane.]]
</td>
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<td>
[[Image:Tempo L7 Fig6.png|thumb|left|550px|Electric field distribution on FSY plane.]]
</td>
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<td>
[[Image:Tempo L7 Fig7.png|thumb|left|550px|Magnetic field distribution on FSY plane.]]
</td>
</tr>
</table>

Visualize the 3D radiation pattern of the waveguide-fed horn antenna in both linear and dB scales. 

<table>
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<td>
[[Image:Tempo L7 Fig8.png|thumb|550px|The 3D radiation pattern of the horn antenna in linear scale.]]
</td>
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<td>
[[Image:Tempo L7 Fig9.png|thumb|550px|The 3D radiation pattern of the horn antenna in dB scale.]]
</td>
</tr>
</table>

From the variables dialog you can see that the value of the variable "gain_dB" was set to 15dB. Also, the aperture dimensions and length of the pyramidal horn section were chosen based on design formulas to yield an antenna gain of 15dB. From the 3D radiation pattern plot, you can see that the computed directivity of your horn structure is 15.02dB. 

== Designing a Long Horn Antenna ==

In this section of the tutorial lesson, you are going to change the aperture dimensions and length of the pyramidal horn section according to the table below:

{| class="wikitable" style="text-align: center;"
|-
! scope="col"| Variable Name
! scope="col"| Original Definition
! scope="col"| New Definition
|-
| horn_aperture_a
| horn_design_a(gain_dB,wg_a/lambda0_unit,wg_b/lambda0_unit)*lambda0_unit 
| 80 
|-
| horn_aperture_b
| horn_design_b(gain_dB,wg_a/lambda0_unit,wg_b/lambda0_unit)*lambda0_unit
| 60 
|-
| horn_len
| horn_design_l(gain_dB,wg_a/lambda0_unit,wg_b/lambda0_unit)*lambda0_unit
| 200
|}

Your horn antenna geometry should now look like this:

<table>
<tr>
<td> [[Image:Tempo L7 Fig12.png|thumb|400px|The geometry of the long horn antenna.]] </td>
</tr>
</table>

After meshing the longer horn, you will see a new number triplet in the status bar: 231 × 185 × 279. The total number of mesh cells can be calculated as (231 -1 + 2×8) × (185 -1 + 2×8) × (279 -1 + 2×8) = 245 × 199 × 295 = 14,464,800 mesh cells. This is a fairly large computational problem. Run an FDTD analysis of the new horn structure and plots its near-field distributions and far-field radiation patterns.  

<table>
<tr>
<td>
[[Image:Tempo_L7_Fig14.png|thumb|left|550px|Electric field distribution of the long horn antenna on FSX plane.]]
</td>
</tr>
<tr>
<td>
[[Image:Tempo_L7_Fig13.png|thumb|left|550px|Magnetic field distribution of the long horn antenna on FSX plane.]]
</td>
</tr>
<tr>
<td>
[[Image:Tempo L7 Fig15.png|thumb|left|550px|Electric field distribution of the long horn antenna on FSY plane.]]
</tr>
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<td>
[[Image:Tempo L7 Fig16.png|thumb|left|550px|Magnetic field distribution of the long horn antenna on FSY plane.]]
</td>
</tr>
</table>

From the radiation pattern plots, you can see that the directivity of the longer horn has increased to 18.74dB.

<table>
<tr>
<td>
[[Image:Tempo L7 Fig17.png|thumb|550px|The 3D radiation pattern of the long horn antenna in linear scale.]]
</td>
</tr>
<tr>
<td>
[[Image:Tempo L7 Fig18.png|thumb|550px|The 3D radiation pattern of the long horn antenna in dB scale.]]
</td>
</tr>
</table>

The figures below show the 2D Cartesian and polar graphs of the radiation patterns in the principal E-plane and H-plane of the horn antenna.

<table>
<tr>
<td>
[[Image:Tempo L7 Fig19.png|thumb|left|480px|The 2D Cartesian graph of radiation pattern of the long horn antenna in the YZ plane.]]
</td>
</tr>
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<td>
[[Image:Tempo L7 Fig20.png|thumb|left|480px|The 2D Cartesian graph of radiation pattern of the long horn antenna in the ZX plane.]]
</td>
</tr>
<tr>
<td>
[[Image:Tempo L7 Fig21.png|thumb|left|480px|The 2D polar graph of radiation pattern of the long horn antenna in the YZ plane.]]
</td>
</tr>
<tr>
<td>
[[Image:Tempo L7 Fig22.png|thumb|left|480px|The 2D polar graph of radiation pattern of the long horn antenna in the ZX plane.]]
</td>
</tr>
</table>

== Verifying Your Simulation Results ==

In order to verify your simulation results, you will consider the approximate analytical expressions for the directivity of a pyramidal horn antenna of aperture dimensions A &times; B and length L fed by a rectangular waveguide of dimensions a &times; b. These are given by: 

<math>  D_P = \frac{4\pi AB}{\lambda_0^2} \epsilon_t \epsilon_E \epsilon_H  </math>

where 

<math>  \epsilon_t = \frac{8}{\pi^2} </math>

<math>  \epsilon_H =  \frac{\pi^2}{64t} \left[ \left[ C(p_1) - C(p_2) \right]^2 +  \left[ S(p_1) - S(p_2) \right]^2  \right] </math>

<math>  \epsilon_E =  \frac{\left[ C(q) \right]^2 +  \left[ S(q) \right]^2}{q^2}   </math>

<math>  t = \frac{1}{8R_{0H}/\lambda_0} \left( \frac{a}{\lambda_0} \right)^2  </math>

<math>  p_1 = 2\sqrt{t} \left( 1+ \frac{1}{8t} \right)  </math>

<math>  p_2 = 2\sqrt{t} \left( -1+ \frac{1}{8t} \right)  </math>

<math>  q = \frac{b}{\sqrt{2\lambda_0 R_{0E}}}   </math>

and the pyramidal radii R<sub>0H</sub> and R<sub>0E</sub> are given by:

<math>  R_{0H} = \frac{L}{1-a/A}   </math>
  
<math>  R_{0E} = \frac{L}{1-b/B}   </math>
  
For your pyramidal horn antenna, A = 80cm, B = 60cm, L = 200cm, a = 16.51cm, b = 8.255cm and &lambda;<sub>0</sub> = 25cm. Therefore, the pyramidal radii are R<sub>0H</sub> = 252.008cm and R<sub>0E</sub> = 231.906cm, and t = 0.005408.

In the above formulas, C(x) and S(x) are the Fresnel cosine and sine integral functions. These are advanced mathematical functions expressed as complicated integrals that can only be evaluated numerically.  [[EM.Cube]] provides a large set of advanced mathematical function as Python libraries or modules. For example, you can find the syntax of the Fresnel integral functions from [[EM.Cube]]'s functions list. To open this list, click the '''Functions''' [[File:Functions icon.png]] button of the Simulate Toolbar or select the menu item '''Simulate &rarr; Functions...'''. 

<table>
<tr>
<td>
[[Image:Tempo L7 Fig25.png|thumb|720px|EM.Cube's Functions List.]]
</td>
</tr>
</table>

Python is a versatile scripting language that can be used very effectively for all kinds of mathematical calculations. You can use [[EM.Cube]]'s '''Python Command Line''' as an online calculator or execute sophisticated Python scripts. The figure below shows some of the above calculations using Python's command line:
 
<table>
<tr>
<td>
[[Image:Tempo L7 Fig27.png|thumb|480px|Performing mathematical calculations using EM.Cube's Python command line.]]
</td>
</tr>
</table>

Based on the above calculations, you will find that p1 = 3.546579, p2 = 3.252418 and q = 0.076661. The approximate analytical value of the directivity of the pyramidal horn antenna is finally found as:

<math>  \epsilon_t = 0.8106, \quad \epsilon_H = 0.999979, \quad \epsilon_E =  0.999992   </math>

<math>  D_p= \frac{(4\pi)(0.8)(0.6)}{(0.25)^2} (0.8106)(0.999979)(0.999992   ) = 78.226 = 18.93dB</math>

which agrees fairly well with the directivity value computed by [[EM.Tempo]].

== Writing a Python Script for Calculation of Pyramidal Horn Directivity ==

At the last step of this tutorial lesson, you will write your own first Python function to calculate the directivity of a pyramidal horn antenna. The input and output of such a function are:

INPUTS: l0 (m), a (m), b (m), A (m), B (m), L (m)

OUTPUT: D0 

You can write a Python script in any text editor outside [[EM.Cube]]. But [[EM.Cube]] also provides a convenient utility for creating and managing Python functions. For this purpose, click the '''Models''' [[File:Models icon.png]] button of the Simulate Toolbar or select the menu item '''Simulate &rarr; Python Models & Scripts...''' to open the Models Dialog.   

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<td>
[[Image:Tempo L7 Fig26.png|thumb|720px|EM.Cube's models dialog.]]
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</table>

This dialog provides a comprehensive list of all the preloaded python functions and scripts as well as your own user-defined Python functions. Select any file from the list and click the {{key|Open}} button of the dialog to open it in a text editor of your choice, which you have associated with the ".PY" file extension in Microsoft Windows. Two common choices are Windows Notepad or Notepad++. 

[[EM.Cube]] also provides a blank Python file called "blank.PY" that is located in the folder "Documents &rarr; EMG &rarr; Python" and can be used as a template for your scripts. Select "blank.PY" from the list and open it. First, save the file as "pyram_horn_d0.PY" to change its name. The portion of the function that you have to enter in the provided template is given below:

------------

def pyram_horn_d0(l0,a,b,A,B,L):

	#Write your python code here

	Roh = L/(1-a/A)

	Roe = L/(1-b/B)

	t = (a/l0)**2/8/(Roh/l0) 

	p1 = 2*sqrt(t)*(1 + 1/8/t)

	p2 = 2*sqrt(t)*(-1 + 1/8/t)

	q = b/sqrt(2*l0*Roe)

	fp1 =  sp.fresnel(p1)

	fp2 =  sp.fresnel(p2)

	fq =  sp.fresnel(q)

	G1 = (fp1[1]-fp2[1])**2 + (fp1[0]-fp2[0])**2

	G2 = abs(fq[1])**2 + abs(fq[0])**2
	et = 8/pi**2

	eh = pi**2*G1/(64*t)

	ee = G2/q**2

	D0 = 4*pi*A*B*et*ee*eh/l0**2	

	return D0	

------------

{{Note|In Python indentation is very important. You can indent using the tab key or using a certain number of spaces, but you cannot mix the two ways in the same script.}}

The figure below shows the entire script in Notepad++.
 
<table>
<tr>
<td>
[[Image:Tempo L7 Fig29.png|thumb|550px|The Python file "pyram_horn_d0.PY" opened in the text editor Notepad++.]]
</td>
</tr>
</table>

Once your Python script is complete, you can load it to [[EM.Cube]] using the {{key|Update}} button of the Models dialog and use it at the command line. If you start typing the name of your new Python function in the command line, a small tip window will appear that helps you remember the syntax of the command. 

{{Note|Oftentimes you might have to exit the [[EM.Cube]] application and open it again to use your new Python functions.}}

<table>
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<td>
[[Image:Tempo L7 Fig30.png|thumb|550px|Typing a new Python command in the command line.]]
</td>
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</table>

If you select the syntax and double-click on the tip window, it will be copied into the command window. Now you can replace the arguments with their numeric values relevant to this project: 

l0 = 25m, a = 0.1651m, b = 0.08255m, A = 0.8m, B = 0.6m and L = 2m 

So, in the command line enter "pyram_horn_d0(0.25,0.1651,0.08255,0.8,0.6,2)" and hit the {{key|Enter}} key of the keyboard. The Python function is evaluated, and the result (D0 = 78.22) is reported on the screen.  

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<td>
[[Image:Tempo L7 Fig31.png|thumb|550px|Executing the new Python command in the command line.]]
</td>
</tr>
</table>

<p>&nbsp;</p>
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