*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 why a capacitive feed point impedance is required for a beta or hairpin matching system, let's break down the concepts:

### 1. Why the Correct Answer is Correct

- **Capacitive Impedance**: A beta or hairpin matching system is used to match the impedance of a Yagi antenna's driven element to the transmission line. When the driven element is electrically shorter than 1/2 wavelength, it exhibits a capacitive reactance. The beta or hairpin match, which is an inductive component, compensates for this capacitive reactance, bringing the impedance to a purely resistive value that matches the transmission line.

### 2. Why the Other Choices are Incorrect

- **Inductive (Choice B)**: If the driven element is electrically longer than 1/2 wavelength, it would exhibit inductive reactance. A beta or hairpin match, which is inductive, cannot effectively match an inductive impedance to a resistive one without additional components.

- **Purely Resistive (Choice C)**: If the impedance were purely resistive, no matching would be needed, as it would already match the transmission line's impedance.

- **Purely Reactive (Choice D)**: This is too vague. While the driven element's impedance is reactive, the specific type (capacitive or inductive) is crucial for determining the appropriate matching system.

### 3. Useful Formulas or Rules of Thumb

- **Reactance of a Capacitive Element**: 
  $$ X_C = \frac{1}{2\pi f C} $$
  where $X_C$ is the capacitive reactance, $f$ is the frequency, and $C$ is the capacitance.

- **Beta or Hairpin Match**: This is essentially an inductor placed across the feed point to cancel out the capacitive reactance, making the overall impedance purely resistive.

In summary, a beta or hairpin matching system is designed to work with a capacitive impedance, which occurs when the driven element is electrically shorter than 1/2 wavelength. This allows the system to effectively match the antenna to the transmission line.