Which of these transmission line impedances would be suitable for constructing a quarter-wave Q-section for matching a 100-ohm feed point impedance to a 50-ohm transmission line?
Why A quarter-wave transformer (Q-section) transforms impedances according to Z0 = sqrt(Z_load x Z_source). Here sqrt(100 x 50) = sqrt(5000) = 70.7 ohms, so the nearest readily available coax, 75-ohm line, is the practical choice. The section must be a quarter wavelength long at the operating frequency, accounting for the line's velocity factor.
Watch out The 50-ohm and 90-ohm values are too far from the 70.7-ohm geometric mean, and 62 ohms is below it; the answer must land just under the midpoint between 50 and 100, which 75-ohm cable does while also being a standard cable type.
Q-section = geometric mean: sqrt(50 x 100) = 70.7, and the coax you can actually buy at that value is 75 ohms.