*Question E9F10:
What impedance does a 1/8-wavelength transmission line present to an RF generator when the line is shorted at the far end?*  

*Correct Answer is  C: An inductive reactance*  

### AI Response: (may contain errors)  

When dealing with transmission lines, the impedance presented to an RF generator depends on the length of the line and the termination at the far end. For a line that is shorted at the far end, the impedance transformation can be understood using the concept of electrical length.

### Explanation:

1. **Correct Answer: An inductive reactance**

   - A 1/8-wavelength transmission line that is shorted at the far end will transform the short circuit into an inductive reactance at the input.
   - The impedance transformation for a shorted line is periodic with a wavelength. At 1/8 wavelength, the line transforms the short (zero impedance) into an inductive reactance.
   - This is because the line is less than a quarter wavelength, and the impedance transformation for a shorted line at less than a quarter wavelength results in inductive behavior.

2. **Why the Other Choices are Incorrect:**

   - **A: A capacitive reactance**  
     - A capacitive reactance would occur if the line were open at the far end and less than a quarter wavelength. Since the line is shorted, this is not the case.
   
   - **B: The same as the characteristic impedance of the line**  
     - The characteristic impedance is only presented when the line is terminated in its characteristic impedance, not when it is shorted.
   
   - **D: Zero**  
     - Zero impedance would be the case if the line were a multiple of a half-wavelength long and shorted. At 1/8 wavelength, the impedance is not zero.

### Useful Formula:

The impedance transformation for a shorted transmission line can be described by the formula:

$$
Z_{\text{in}} = j Z_0 \tan\left(\frac{2\pi \ell}{\lambda}\right)
$$

where:
- $Z_{\text{in}}$ is the input impedance.
- $Z_0$ is the characteristic impedance of the line.
- $\ell$ is the physical length of the line.
- $\lambda$ is the wavelength of the signal.
- $j$ indicates the imaginary unit, representing reactance.

For a 1/8-wavelength line, $\frac{2\pi \ell}{\lambda} = \frac{\pi}{4}$, leading to an inductive reactance.