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[ 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 — and that’s the “good story” of how a simple trigonometric insight saves your antenna from becoming a dummy load.

To find ( y_0 ) for ( Z_{in} = 50 \ \Omega ):

She laughed — a tired, relieved laugh. The calculator hadn’t lied. The cosine-squared impedance taper worked.

That’s where the “inset feed calculator” entered — not as a fancy app, but as a haunting set of equations.