1. Hanyang University 1/14 Antennas & RF Devices Lab. MODERN ANTENNA HANDBOOK by CONSTANTINE A.BALANIS ch. 5.2 – 5.2.2 Kim Sung Peel

2. Hanyang University 2/14 Contents 5.2 Radiation Pattern Analysis - 5.2.1 Dual-Reflector Systems - 5.2.2 Useful Coordinate Transformations

3. Hanyang University 3/14 5.2 Radiation Pattern Analysis Physical optics(PO): most important analytical tool for radiation pattern analysis. →It is used to calculate the scattered field from a metallic reflecting surface(reflector antenna). Electrical currents, which excite the scattered field, are induced on the conducting surface by an incident wave. → PO surface current( ) on the subreflector surface: ( )

4. Hanyang University 4/14 5.2 Radiation Pattern Analysis Discrete approximation of the radiation integral: one of the simplest possible calculation. →Replace the actual reflector surface measurements with a triangular facet representation, so that the reflector resembles a geodesic dome.

5. Hanyang University 5/14 5.2 Radiation Pattern Analysis The PO current is assumed to be constant in magnitude and phase over each facet, so the radiation integral is reduced to a simple summation. The PO radiation integral over the reflector surface( ): ( ) Purpose of analysis: the true surface( ) is replaced by a contiguous set of triangular facets. These facets ( ) are chosen to be equal in size. ∴ The field obtained by replacing the true surface( ) with the set of triangular facet: ( )

6. Hanyang University 6/14 5.2 Radiation Pattern Analysis Assume that the necessary transformations have been performed so that the incident field is given in terms of the reflector coordinate system. Applying Eq. (5.6) and (5.7) to Eq. (5.5). Then (5.5) can be written as (5.8):

7. Hanyang University 7/14 5.2 Radiation Pattern Analysis ( ) And the Jacobian is Making use of the Jacobian then allow Eq. (5.8) to be rewritten as (In which represents the area of the i th triangular facet projected onto the z = 0 plane.) Now, make a Taylor-series expansion of the exponent in upper equation. Retaining only the first-order terms, one can formally write

8. Hanyang University 8/14 5.2 Radiation Pattern Analysis It may be observed that this integral is the 2D Fourier transform of the i th projected triangle ∴ Full radiation integral = Sum of all the transforms of the individual triangles. →2D Fourier transform of the i th projected triangle Final form of the linear-phase approximation over each triangular facet:

9. Hanyang University 9/14 5.2.1 Dual-Reflector Systems ① Feed illuminates the subreflector, and the currents on the subreflector surface are determined. ② Scattered field from the subreflector are used to illuminate the main reflector, and its induced currents are determined. ③ The main reflector’s scattered fields are determined by integrating these currents. Many coordinate systems are required to allow flexibility in locating and orienting the feed, subreflector, main reflector, and output-pattern generation PO integration methodology is sequential fashion for analyzing the dual-reflector system.

10. Hanyang University 10/14 5.2.2 Useful Coordinate Transformations In general, the radiated field of feed coordinates do not coincide with the reflector coordinates, therefore a coordinate transformation is needed to relate the two systems. Cartesian-to-spherical transformation: a vector field H can be expressed by its Cartesian or spherical components.

11. Hanyang University 11/14 5.2.2 Useful Coordinate Transformations Cartesian-to-Cartesian transformation:

12. Hanyang University 12/14 5.2.2 Useful Coordinate Transformations Cartesian-to-Cartesian transformation:

13. Hanyang University 13/14 Thank you for your attention