Question E9C01:
What type of radiation pattern is created by two 1/4-wavelength vertical antennas spaced 1/2-wavelength apart and fed 180 degrees out of phase?  

Choices:  
A: Cardioid  
B: Omni-directional  
C: A figure-eight broadside to the axis of the array  
D: A figure-eight oriented along the axis of the array    

Correct Answer is  D: A figure-eight oriented along the axis of the array  

Two 1/4-wave verticals spaced 1/2λ and fed 180° out of phase form a two-element endfire array in azimuth. The maximum radiation is along the line connecting the two antennas (the array axis), producing a figure-eight pattern oriented along that axis.

Why D is correct (figure-eight along the axis)
- The two-element array factor for elements spaced distance $d$ and phase difference $\beta$ is:
  $$AF(\phi) = 2\cos\!\left(\frac{kd\cos\phi + \beta}{2}\right), \quad k=\frac{2\pi}{\lambda}$$
- With $d = \tfrac{\lambda}{2} \Rightarrow kd = \pi$ and $\beta = \pi$ (180° out of phase):
  $$AF(\phi) = 2\cos\!\left(\frac{\pi\cos\phi + \pi}{2}\right)$$
- Maxima occur when the cosine’s argument equals $m\pi$:
  $$\pi\cos\phi + \pi = 2m\pi \Rightarrow \cos\phi = 2m - 1$$
  Valid solutions: $\phi = 0^\circ, 180^\circ$ (endfire, along the array axis).
- Nulls occur broadside ($\phi = 90^\circ$), yielding a classic figure-eight aligned with the axis.

Why the other choices are incorrect
- A: Cardioid — Requires roughly $d \approx \tfrac{\lambda}{4}$ and a $90^\circ$ phase shift (or suitable amplitude/phase taper) to create a single deep null. Here $d=\tfrac{\lambda}{2}$ and $\beta=180^\circ$, so not cardioid.
- B: Omni-directional — A single vertical is omni in azimuth, but two spaced and phased elements produce interference lobes and nulls; spacing of $0.5\lambda$ with 180° phasing is far from omni.
- C: Figure-eight broadside — That occurs with $d=\tfrac{\lambda}{2}$ and in-phase feeding ($\beta=0$). Our array is out of phase, so the figure-eight rotates to endfire (along the axis), not broadside.

Useful rules of thumb
- Two elements, $d=\tfrac{\lambda}{2}$:
  - In-phase ($\beta=0$) → broadside figure-eight (max perpendicular to axis).
  - Out of phase ($\beta=\pi$) → endfire figure-eight (max along axis).
- Cardioid pattern commonly: $d\approx \tfrac{\lambda}{4}$ with $\beta \approx 90^\circ$.