Question E9E06:
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?  

Choices:  
A: 50 ohms  
B: 62 ohms  
C: 75 ohms  
D: 90 ohms    

Correct Answer is  C: 75 ohms  

To match 100 Ω to a 50 Ω line with a quarter-wave transformer (Q-section), use the geometric mean rule.

Key formula (lossless quarter-wave section):
- Input impedance looking into the λ/4 section:  
  $$Z_\text{in} = \frac{Z_0^2}{Z_L}$$
- For a match, set $Z_\text{in} = Z_S$ and solve for $Z_0$:  
  $$Z_0 = \sqrt{Z_S Z_L}$$

Apply to this problem:
- $Z_S = 50\ \Omega$, $Z_L = 100\ \Omega$  
  $$Z_0 = \sqrt{50 \times 100} = \sqrt{5000} \approx 70.7\ \Omega$$
- The closest standard impedance is 75 Ω, so 75 Ω is suitable.

Why 75 Ω (C) is correct:
- 75 Ω is the nearest standard to the ideal 70.7 Ω, giving a good match for a λ/4 transformer between 50 Ω and 100 Ω.

Why the others are incorrect (using $Z_\text{in} = Z_0^2 / Z_L$):
- 50 Ω: $$Z_\text{in} = \frac{50^2}{100} = 25\ \Omega$$ Not 50 Ω, poor match.
- 62 Ω: $$Z_\text{in} = \frac{62^2}{100} = 38.4\ \Omega$$ Not 50 Ω.
- 90 Ω: $$Z_\text{in} = \frac{90^2}{100} = 81\ \Omega$$ Not 50 Ω.

Rules of thumb:
- Quarter-wave transformer characteristic impedance:  
  $$Z_0 = \sqrt{Z_\text{line} \cdot Z_\text{load}}$$
- The section must be an electrical quarter wavelength (or any odd multiple):  
  $$\ell = \frac{\lambda}{4} \times \text{VF}$$
  where VF is the velocity factor of the line.