# E and H pattern of 2 element antenna array.

This is not a homework, too old for that!!! I just have a question that I create myself. All the books only show the pattern that is more obvious.......they show either the E or the H pattern. I took an exercise that asked for the H pattern, in turn, using the pattern multiplication to try to find the E pattern and ran into road block. Here is the exercise:

Given two Hertzian dipoles oriented in z-direction. Both line up on x-axis and $\;d=\frac {\lambda} 2 \;$ apart. Both are driven by the same amplitude and phase $\alpha =0$. Find the E and H pattern.

From pattern multiplication:

$$|E(\theta, \phi)| = \frac {E_m}{R_0}\;| F(\theta, \phi)|\;\left|\cos\left(\frac {\beta d \cos\phi \sin\theta -\alpha} {2} \right)\right|$$

Where $\;| F(\theta, \phi)|= |\sin\theta| \;$ is the element factor for the Hertzian dipole of each element and $\;\left|\cos\left(\frac {\beta d \cos\phi \sin\theta -\alpha} 2 \right)\right|$ is the array factor.

The pattern function is:

$$| F(\theta, \phi)|\;\left|\cos\left(\frac {\beta d \cos\phi \sin\theta -\alpha} {2} \right)\right|$$

I have no problem getting the H pattern by just putting $\;\theta=\frac {\pi}{2}$. I get the two almost ball shape one on +ve y axis and one on -ve y axis. There are no E field on x direction as expected.

But when I try to look at the E pattern at $\;\phi=0$, I don't get what I expected. From the H pattern above, I expect I'll get no E field at $\;\phi=0 \;\hbox { and } \phi=\pi$ for all angle of $\;\theta$. But according to the pattern function:

$$| F(\theta, \phi)|\;\left|\cos\left(\frac {\beta d \cos\phi \sin\theta -\alpha} {2} \right)\right| = |\sin\theta| \left|\cos\left(\frac {\pi}{2}( \sin\theta ) \right)\right|$$

You can see it is zero when $\theta= 0 \;\hbox { or }\;\theta=\frac{\pi}{2}$, but it is not zero in between. Can anyone help explaining this?

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Anyone?

I know the equations are correct as it is given in the book and was used to solve the H pattern. Far as my understanding, array pattern is not E or H pattern dependent, in fact it is not antenna elements dependent. It only depend on the $\theta, \phi, \alpha, d$.