Microstrip and Printed Antenna Design
Written for practicing engineers new to the field, this book includes a succinct treatment of wireless designs with enough detail that CAD formulas may be implemented with relative ease.
TABLE OF CONTENTS Microstrip and Printed Antenna Design
Appendix B: Numerical Methods
B.1 Numerical Integration
The integrations presented in Chapter 2 to compute the directivity of a rectangular microstrip antenna (2.44), and the integrations in Chapter 3 to compute the radiation conductance and radiation Q of a circular microstrip antenna (3.9) have no known closed form solution and are evaluated numerically.
When an integrand is sufficiently smooth and contains no singularities, a very efficient method of numerical integration is Gaussian Quadrature. Gaussian Quadrature integrates a polynomial which curve fits the integrand. The integrations presented in this book all have well behaved integrands and may be integrated using this method. The integrations of interest are of this form: [1]
Where N is the number of Gaussian quadrature points, W i are the weights and Z i are the abscissas.
For a 16 point Gaussian quadrature the sums are evaluated as:
and
The Gaussian weights W i and abscissas Z i for a 16 point Gaussian quadrature are presented below:
| Z i | W i |
|---|---|
| 0.09501 25098 37637 440185 | 0.18945 06104 55068 496285 |
| 0.28160 35507 79258 913230 | 0.18260 34150 44923 588867 |
| 0.45801 67776 57227 386342 | 0.16915 65193 95002 538189 |
| 0.61787 62444 02643 748447 | 0.14959 59888 16576 732081 |
| 0.75540 44083 55003 033895 | 0.12462 89712 55533 872052 |
| 0.86563 12023 87831 743880 | 0.09515 85116 82492 784810 |
| 0.94457 50230 73232 576078 | 0.06225 35239 38647 892863 |
| 0.98940 09349 91649 932596 | 0.02715 24594 11754 094852 |
The integrations in this text are all calculated using a 96 point Gaussian Quadrature. [2]
[1]Hewlett-Packard, HP-41...
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