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Patch antenna Transmission-Line-Model impedance

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bya036

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Why does the impedance at the patch input is purely real ?
As we all know the patch antenna input impedance stems from the transformation of the same impedance from one end of the patch to the other.
As balanis says the imaginary part of the addmitance in one side cancels with the imaginary part of the other side when transformed through the patch which is half wavelength long. But it is evident that half wavelngth transformation do a complete circle around the smith center which means no change in impedance which means that instead canceling the impedance will add - both imaginary and real ?
Please help!
 

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