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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 [itex]\;d=\frac {\lambda} 2 \;[/itex] apart. Both are driven by the same amplitude and phase [itex]\alpha =0[/itex]. Find the E and H pattern.

From pattern multiplication:

[tex]|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|[/tex]

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

The pattern function is:

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

I have no problem getting the H pattern by just putting [itex]\;\theta=\frac {\pi}{2}[/itex]. 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 [itex]\;\phi=0[/itex], I don't get what I expected. From the H pattern above, I expect I'll get no E field at [itex]\;\phi=0 \;\hbox { and } \phi=\pi[/itex] for all angle of [itex]\;\theta[/itex]. But according to the pattern function:

[tex]| 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|[/tex]

You can see it is zero when [itex]\theta= 0 \;\hbox { or }\;\theta=\frac{\pi}{2}[/itex], 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 [itex]\;d=\frac {\lambda} 2 \;[/itex] apart. Both are driven by the same amplitude and phase [itex]\alpha =0[/itex]. Find the E and H pattern.

From pattern multiplication:

[tex]|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|[/tex]

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

The pattern function is:

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

I have no problem getting the H pattern by just putting [itex]\;\theta=\frac {\pi}{2}[/itex]. 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 [itex]\;\phi=0[/itex], I don't get what I expected. From the H pattern above, I expect I'll get no E field at [itex]\;\phi=0 \;\hbox { and } \phi=\pi[/itex] for all angle of [itex]\;\theta[/itex]. But according to the pattern function:

[tex]| 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|[/tex]

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

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