*Question E9E05:
What Yagi driven element feed point impedance is required to use a beta or hairpin matching system?*  

*Correct Answer is  A: Capacitive (driven element electrically shorter than 1/2 wavelength)*  

### AI Response: (may contain errors)  

To understand the correct answer for the Yagi driven element feed point impedance required for a beta or hairpin matching system, let's break down the concepts involved.

### Correct Answer: A - Capacitive (driven element electrically shorter than 1/2 wavelength)

1. **Why the correct answer is correct:**
   - A beta or hairpin matching system is used to match the impedance of a Yagi antenna's driven element to the feed line. When the driven element is electrically shorter than 1/2 wavelength, it presents a capacitive reactance at the feed point. This means that the impedance is not purely resistive and has a negative reactance (capacitive).
   - The beta match is designed to cancel this capacitive reactance, allowing for a better match to the 50-ohm feed line.

### Why the other choices are incorrect:

2. **B - Inductive (driven element electrically longer than 1/2 wavelength):**
   - If the driven element is longer than 1/2 wavelength, it presents an inductive reactance at the feed point. This is the opposite of what is needed for a beta match, which is designed for capacitive elements.

3. **C - Purely resistive:**
   - A purely resistive impedance would imply that there is no reactance present. However, for a driven element that is shorter than 1/2 wavelength, there will always be some capacitive reactance, making this choice incorrect.

4. **D - Purely reactive:**
   - A purely reactive impedance means that the impedance consists only of reactance (either capacitive or inductive) with no resistive component. Since we are dealing with a driven element that is shorter than 1/2 wavelength, it will have a capacitive reactance, but it will also have a resistive component due to the radiation resistance of the antenna. Therefore, this choice is also incorrect.

### Useful Formulas and Rules of Thumb:

- The impedance of a dipole antenna can be approximated using the formula:
  $$ Z = R + jX $$
  where:
  - $R$ = radiation resistance (typically around 73 ohms for a half-wave dipole)
  - $X$ = reactance (negative for capacitive, positive for inductive)

- For a driven element shorter than 1/2 wavelength, the reactance $X$ is negative (capacitive), which is what the beta match compensates for.

In summary, the beta or hairpin matching system is specifically designed to work with driven elements that are electrically shorter than 1/2 wavelength, making the correct answer A - Capacitive.