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yungman
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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?
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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