Question E9C02:
What type of radiation pattern is created by two 1/4-wavelength vertical antennas spaced 1/4-wavelength apart and fed 90 degrees out of phase?  

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

Correct Answer is  A: Cardioid  

To get the pattern of a two-element phased vertical array, use the two-source array factor:
$$AF(\theta) = 1 + e^{j\,(k d \cos\theta + \alpha)}$$
where:
- $k = 2\pi/\lambda$
- $d$ = spacing between elements
- $\alpha$ = phase difference between element currents (second relative to first)
- $\theta$ is measured from the axis of the array (end-fire direction)

Given $d = \lambda/4$ and $\alpha = 90^\circ = \pi/2$, we have $k d = \pi/2$, so:
- Nulls occur when $k d \cos\theta + \alpha = (2m+1)\pi$. With $m=0$:
  $$(\pi/2)\cos\theta + \pi/2 = \pi \Rightarrow \cos\theta = 1 \Rightarrow \theta = 0^\circ$$
- Maxima occur when $k d \cos\theta + \alpha = 2m\pi$. With $m=0$:
  $$(\pi/2)\cos\theta + \pi/2 = 0 \Rightarrow \cos\theta = -1 \Rightarrow \theta = 180^\circ$$

This produces a single main lobe in one end-fire direction and a deep null in the opposite direction: a cardioid pattern.

Why A (Cardioid) is correct:
- Quarter-wave spacing ($\lambda/4$) with quadrature phasing ($90^\circ$) is the classic condition for a unidirectional cardioid in the horizontal plane with two verticals.

Why the others are incorrect:
- B: Figure-eight end-fire requires equal amplitudes with about $180^\circ$ phasing and small spacing; it yields two opposite end-fire lobes, not a single-lobed cardioid.
- C: Figure-eight broadside occurs with in-phase feeding ($\alpha=0$) and spacing near $\lambda/2$, giving maxima perpendicular to the array axis.
- D: Omni-directional is what a single vertical has; two spaced, phased elements do not produce an omni pattern.

Rules of thumb:
- $d \approx \lambda/2$, $\alpha = 0^\circ$ → broadside figure-eight.
- Small $d$, $\alpha \approx 180^\circ$ → bidirectional end-fire figure-eight.
- $d \approx \lambda/4$, $\alpha \approx \pm 90^\circ$ → cardioid (unidirectional end-fire). The main lobe points toward one element and a deep null toward the other, depending on which element leads in phase.