Most online calculators just solve this iteratively — and that’s the “good story” of how a simple trigonometric insight saves your antenna from becoming a dummy load.
Three days later, the etched board sat on the VNA. She pressed the SMA connector gently against the inset feed point. The display flickered… then locked. inset fed microstrip patch antenna calculator
[ Z_{in}(y=y_0) = Z_{edge} \cdot \cos^2\left( \frac{\pi y_0}{L} \right) ] where [ Z_{edge} \approx 90 \cdot \frac{\varepsilon_r^2}{\varepsilon_r - 1} \left( \frac{L}{W} \right) ] (for narrow patches; more accurate models use transmission line or cavity methods). Most online calculators just solve this iteratively —
To find ( y_0 ) for ( Z_{in} = 50 \ \Omega ): inset fed microstrip patch antenna calculator