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

Choices:  
A: Capacitive (driven element electrically shorter than 1/2 wavelength)  
B: Inductive (driven element electrically longer than 1/2 wavelength)  
C: Purely resistive  
D: Purely reactive    

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

To use a beta (hairpin) match, the driven element must be slightly shorter than resonance so its feed point is capacitive. The hairpin is a shunt inductor across the feedpoint that cancels this capacitive reactance and steps the resistance up toward the feed line impedance (often 50 Ω).

Why A is correct (Capacitive):
- The hairpin is an inductor in parallel with the feedpoint. It provides positive reactance to cancel the negative (capacitive) reactance of a slightly short element.
- Shorter-than-½λ elements are capacitive: if the driven element is made a bit short, its input reactance is $X_{in} < 0$. The hairpin adds $+jX$ so the net susceptance goes to zero (resonance), while also transforming the resistive part upward.
- In practice, Yagi driven elements often have $R_{in} < 50\ \Omega$; the beta match (capacitive element + shunt inductor) is an L-network that matches $R_{in}$ to $Z_0$.

Why the others are incorrect:
- B: Inductive (element longer than ½λ) — An inductive element ($X_{in} > 0$) needs capacitive compensation, not an inductive hairpin. A shunt inductor would worsen the mismatch.
- C: Purely resistive — If the feedpoint is already resistive (at resonance), there’s no reactance for a hairpin to cancel; a beta match specifically relies on the element being capacitive.
- D: Purely reactive — Too vague. The hairpin requires specifically capacitive reactance to be canceled by the shunt inductor.

Useful formulas and rules of thumb:
- Shorter than resonant length → capacitive; longer → inductive.
- Hairpin (shorted stub) input reactance:
  $$X_{\text{stub}} = Z_0 \tan\!\left(\frac{2\pi l}{\lambda}\right)$$
  For a shorted stub with $0 < l < \lambda/4$, $X_{\text{stub}} > 0$ (inductive).
- Matching condition (conceptual): choose the hairpin so its inductive susceptance cancels the driven element’s capacitive susceptance at the feedpoint.
- L-network rule (when $R_{\text{ant}} < Z_0$): a shunt inductor with a series capacitive reactance gives a match. In a beta match, the “series C” is provided by the short (capacitive) driven element itself.