Mathematica Examples

## B. Analysis of Elementary Antenna Using the Basic Interface.

### Example B.1: Arbitrary Three-Dimensional Array Demo.

An Array Factor (AF) describes the pattern generated by a collection of isotropic point sources. The following illustrates the use of the AF3Ddemo function (defined in the ArrPckdemo.m package). AF3Ddemo is self-conatined and takes an array geometry along with the relative magnitude and phase of the currents that are fed into each element and displays the AF formula, the array structure and the array factor pattern. The following specifies that AF3Ddemo should be executed for a uniformly excited three-element linear array with a spacing of three-quarters of a wavelength between adjacent elements.

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```  The Array Factor Function:
Abs[10 + 20 (Cos[Pi Cos[theta] +
Pi Cos[phi] Sin[theta] + Pi Sin[phi] Sin[theta]]\
- I Sin[Pi Cos[theta] + Pi Cos[phi] Sin[theta] +
Pi Sin[phi] Sin[theta]]) +
20 (Cos[Pi Cos[theta] + Pi Cos[phi] Sin[theta] +
Pi Sin[phi] Sin[theta]] +
I Sin[Pi Cos[theta] + Pi Cos[phi] Sin[theta] +
Pi Sin[phi] Sin[theta]])]

```

 This script was developed at the Center for Computational Electromagnetics (CCEM) in the University of Illinois at Urbana with funding from the Sloan Center for Asynchronous Learning Environments(SCALE).

### Example B.2: Arbitrary Dipole Antennas.

The following illustrates the process of obtaining the far-field radiation patterns of a 1 meter verticle dipole excited at 600 MHz. The PatPlot.m package is loaded to gain access to the PatPlotAll routine which renders the patterns. The PatDip function, defined in the PatDip.m package, is used to specify the radiation pattern of a finite length dipole.

 -UnixWebMath-

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 Mathematica Output
`1/2 LAMBDA DIPOLE PATTERN `
 Processed by Mathematica version 2.2 running on a UNIX platform. This script was developed at the Center for Computational Electromagnetics (CCEM) - University of Illinois at Urbana with funding from the Sloan Center for Asynchronous Learning Environments(SCALE)

### Example B.3: Arbitrary Three-Dimensional Array.

An Array Factor (AF) describes the pattern generated by a collection of isotropic point sources. The following illustrates the use of the AF3D function (defined in the ArrPck.m package) in conjunction with the aforementioned PatPlotAll function. AF3D takes an array geometry along with the relative magnitude and phase of the currents that are fed into each element and returns the AF formula. The following shows the AF pattern of a uniformly excited three-element linear array with a spacing of three-quarters of a wavelength between adjacent elements.

 -UnixWebMath-

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 Mathematica Output
`3 ELEMENT LINEAR ARRAY WITH 3/4 LAMBDA SPACING `
 Processed by Mathematica version 2.2 running on a UNIX platform. This script was developed at the Center for Computational Electromagnetics (CCEM) - University of Illinois at Urbana with funding from the Sloan Center for Asynchronous Learning Environments(SCALE)

### Example B.4: Arbitrary Array of Identical Dipoles.

The radiation pattern of an array of identical elements can be decomposed into two seperate functions: (1) an element pattern and (2) the array factor pattern. As the name implies, the element pattern describes the influence of the antenna type on the fields radiated by the array. The array factor describes the effect of the array structure on the field. The complete pattern is the product of these two functions. Since the PatDip and AF3D functions (illustrated above) both return a mathematical description of their respective patterns, they can be used together to analyze an array of dipoles.

 -UnixWebMath-

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 Mathematica Output
`1/2 LAMBDA DIPOLE PATTERN `
`3 ELEMENT LINEAR ARRAY WITH 3/4 LAMBDA SPACING `
`ARRAY OF DIPOLES`
 Processed by Mathematica version 2.2 running on a UNIX platform. This script was developed at the Center for Computational Electromagnetics (CCEM) - University of Illinois at Urbana with funding from the Sloan Center for Asynchronous Learning Environments(SCALE)

### Example B.5: Design of Optimally Directive Arrays.

Carefully feeding and array can result in highly directive radiation patterns that can be "steered" into a desired direction. Beam steering is accomplished by changing the relative phasing of adjacent elements. Optimization of the current distribution to provide the maximum directivity in a given direction is a more involved problem. Moreover, the optimal current distribution generally corresponds to a superdirective array characterized by an impractical Q-factor. Maximizing the directivity while constraining the value of the Q-factor is significantly more challenging. The following demonstrates the use of one of the packages included on the AWS for performing directivity optimization on an aribitrary planar array. The following input (i) describes an 8-element array in the x-y plane, (ii) specifies the direction for which directivity is to be maximized (iii) specifies that the resulting feed current distribution should correspond to a Q-factor no greater than 2.0 and, (iv) 20 digits of precision should be maintained throughout the analysis. Note that the condition number of the optimization problem is returned as output, so that the user can determine if more precision is required.

 -UnixWebMath-

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 Mathematica Output
```Condition number for this array problem: 17.820879367577734686

Current values for optimal Directivity with Q = 2.:

0.6264502767177395 - 0.03645915531027452 I
-0.588624835347133 - 0.5785366889814525 I
-0.03158264246083917 - 0.3079834077684028 I
0.4420188841759283 - 0.1572778176768101 I
-0.228570527263726 - 0.06083841512181934 I
0.1907858804043586 - 0.4783748800693714 I
0.861098309783999 + 0.1766758942804685 I
-0.4304027379074689 - 0.2574817653901749 I

```

```Directivity is: 3.80875
```

rendition of:

rendition of:

 Processed by Mathematica version 2.2 running on a UNIX platform. This script was developed at the Center for Computational Electromagnetics (CCEM) - University of Illinois at Urbana with funding from the Sloan Center for Asynchronous Learning Environments(SCALE)

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