Is there any mathematical expression which relates antenna performance with probe feed position ?
The position of probe feed alters the impedance matching and electrical length. Then obviously the performance like S11, gain and BW will change.......
Depending on probe position, the value of input impedance changes. As you did not mention the shape of the patch antenna, it is difficult to give you proper answer.
1. For rectangular, circular and euilateral shaped patch antenna, if we move the probe along its symmetry axis, the value of Zin decreases from 1600 to 0 and then increases at fundamental mode. For immediate higher order mode, it decreases from 1400 to 0 and increases 0 to 200 and again decreases. Similar phenomenon is also found for other higher order modes also
2. for asymmetrical antenna, it is difficult to give answer. As a thumb rule, you can use this: Zin decreases from high value to 0 if we move from its surfaces (edge) to its center
Micro strip antenna centre is zero energy point, in turn, zero impedance. Edge is maximum impedance point. Maximum value vary about 15 to 20 % shape to shape. Ideal point of feed is (in consideration with shape) Balancing of field distribution contour and roughly distance proportion between maximum point to centre (Zero). You may not have Accurate mathematics. My experience with circular shape is, one third radius (distance) from edge to centre is feed point.
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