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I have been trying to match my NFC reader coil of size 20*20 cm with a three element circuit - one capacitor in series, another in parallel and one resistor in parallel. I have been using Texas Instruments' method to match the coil. The procedure I'm using is described in this link

I'm getting a reasonable Q of 25, but the coil is not exactly matched to 13.56 MHz. I'm able to match it to 1 MHz give or take from 13.56 MHz. I think I might not have accounted for parasitic elements in the circuit. While following the procedure mentioned above, the calculated impedances and the ones mentioned in the VNA differ by a lot. I have tried to populate the PCB with through hole and SMD components with the latter giving slightly better Q. Can someone help me understand how to get close to the calculated results and how to set up the coil without too much effect of loose contacts or parasitics? Thank you!

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To begin with, your Q is too high. Your passband shape will cause too much sideband attenuation. NFC requires a 2 MHz passband so your Q should be no larger than 7. That is to say that your should not improve the Q as your title suggests but rather that you should make it "worse".

Without knowing the specific circuit values ($C_s$, $C_p$, and $R_p$), it is difficult to further analyze your circuit. I recommend that you edit your question to reflect the aforementioned values if you would like further guidance. Also report the measured complex impedance of your antenna without the matching network installed. A picture of your assembly (with a scale in view) along with pictures of your PCB layers would also be helpful.

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