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If the problem is still valid, I suggest a disk-cone antenna. You can find a lot of information on the internet, e.g.
https://www.youtube.com/watch?v=IaFSX7ycj0I
or different construction https://www.youtube.com/watch?v=xhW4QQruBTo
Some results of my calculation
I understand that the asymmetry of the radiation characteristics (in both planes) is doubtful. I think that the measurement results are correct, but the antenna is asymmetrical in both planes (for example, different screen dimensions).
There is some doubt that the width of the beam is the same...
I suggest literature
1. Kin-Lu Wong Planar Antennas for Wireless Communications
Willey-Interscience 2003
2. Girish Kumar , K. P. Ray, Broadband Microstrip Antennas ,
2003, Artech House
and you can also look at the page where the PIFA antenna examples are shown.Press the "Select...
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