Question E9F04:
What impedance does a 1/2-wavelength transmission line present to an RF generator when the line is shorted at the far end?  

Choices:  
A: Very high impedance  
B: Very low impedance  
C: The same as the characteristic impedance of the line  
D: The same as the output impedance of the RF generator    

Correct Answer is  B: Very low impedance  

To analyze a shorted 1/2-wavelength transmission line, use the input impedance of a lossless line:

$$
Z_\text{in} = Z_0 \,\frac{Z_L + j Z_0 \tan(\beta l)}{Z_0 + j Z_L \tan(\beta l)}
$$

- For a line length $l = n\lambda/2$, $\tan(\beta l) = \tan(n\pi) = 0$, so:
  $$
  Z_\text{in} = Z_L
  $$
- With a short at the load, $Z_L = 0$, so $Z_\text{in} = 0$ → very low impedance.

Why the correct answer is correct (B):
- A 1/2-wavelength line “repeats” the load. A shorted load remains a short at the input, so the generator sees a very low impedance.

Why the other choices are incorrect:
- A: Very high impedance — This occurs with a shorted load only at an odd multiple of quarter wavelengths: $l = (2n+1)\lambda/4$, where $\tan(\beta l) \to \infty$ and $Z_\text{in} = Z_0^2/Z_L \to \infty$ for $Z_L = 0$.
- C: Same as $Z_0$ — The input equals $Z_0$ only when the line is terminated in $Z_0$ (matched load), not when shorted.
- D: Same as the generator’s output impedance — The input impedance is set by the line and its termination, independent of the generator’s impedance.

Rules of thumb:
- Half-wave line: $l = n\lambda/2 \Rightarrow Z_\text{in} = Z_L$ (load is repeated).
- Quarter-wave line: $l = (2n+1)\lambda/4 \Rightarrow Z_\text{in} = Z_0^2 / Z_L$ (impedance is inverted).