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Why, for realistic directional antennas, are its nulls sharper (deeper, narrower, etc.), than the radiation pattern’s lobes (local angular maxima)? Is there some physics or mathematical geometry to EM fields that requires this to be true?

Or is there a some way to do the opposite, make an antenna with sharper (narrower) lobes than nulls?

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    $\begingroup$ The hairy ball theorem explains why there must be a null, but not why the lobes can't be sharp. $\endgroup$ Jul 20 at 16:16
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    $\begingroup$ Based on the answers so far, it might be worthwhile to clarify what you mean by "sharp". Do you mean "narrow"? Or "pointy"? $\endgroup$ Jul 20 at 21:34
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Its all in the nature of nulls:

  • Two signal sources (same frequency) can possibly subtract exactly producing a very deep null.

  • Two signal sources (same frequency) can only add constructively to double amplitude, at best.

Or, if you wish, imagine a sinusoidal wave whose average value is 1.0, and whose peak is also 1.0. I've chosen a frequency of 1 Hz, but that's not really important.

  • The sinusoid adds the 1.0 DC to the 1.0 peak to produce a peak of 2.0.
  • The 1.0 DC subtracts from the other peak to produce exactly 0.0.


Even though the subtraction goes to zero very smoothly, the approach to zero is very sharp, which can be seen on a log plot of the waveform. The approach to the 2.0 peak is certainly not-so-sharp.

sin(1wt)+1.0 plotted with log scale

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    $\begingroup$ This explains why lobes can't be "strong", but is that the same thing as "sharp"? To me, "sharp" suggests a discontinuity in the first derivative. Why can't we have sharp lobes at 1 dB? $\endgroup$ Jul 20 at 18:42
  • $\begingroup$ @PhilFrost-W8II Yes, nulls tend toward that discontinuity. Think of a 4-element bridge, which produces a null at its detector terminals. A perfect null is extremely sharp and corresponds to perfect bridge balance...the math gives a ratio whose denominator is zero - a point of infinite slope switching from +ve to -ve.,,,a discontinuity. $\endgroup$
    – glen_geek
    Jul 20 at 19:35
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    $\begingroup$ Yes that's true, but I don't think you've explained why, which is what the question is asking. Why can't we have a pattern which looks like your picture flipped vertically, with the peaks at some finite and reasonable value, and the minimum values at 0? $\endgroup$ Jul 20 at 20:52
  • $\begingroup$ @PhilFrost-W8II Your scenario CAN happen if you have a gain element. Armstrong did it with his regenerative amplifier. A 50 ohm resonant antenna can generate a very large voltage if you apply a negative conjugate impedance. The amplifier gain requirement to generate the negative impedance is really quite modest, but the antenna impedance must be exactly matched. And any detector you add to extract that boosted signal upsets the match. Still not quite the OP's scenario though, where all antenna elements have positive impedance magnitudes. $\endgroup$
    – glen_geek
    Jul 20 at 22:45
  • $\begingroup$ Well yes, we can have resonant systems with a sharp peak in the frequency response, but the question is asking about antenna radiation patterns. I can't articulate exactly why, but I don't think I've ever seen a lobe in a radiation pattern with a discontinuous derivative. $\endgroup$ Jul 21 at 2:41
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To increase an antenna's gain, the elements of the antenna must be arranged such that they interfere constructively in the intended direction, and destructively in every other direction. This is possible because varying the distance to an element varies the phase of the wave, so in some positions the phase difference is such that the waves interfere constructively, while in other positions the waves interfere destructively. Most positions experience some interference somewhere between these two extremes.

Now let's consider the math of wave interference a bit with a simple example. Wikipedia has the full derivation, but the tl;dr is the addition of two identical waves differing only in phase by $\varphi$ is:

$$ \underbrace{A\cos(kx-\omega t)}_\text{wave 1} + \underbrace{A\cos(kx-\omega t+\varphi) }_\text{wave 2} = \underbrace{2A\cos \left( \varphi \over 2 \right)}_\text{amplitude} \cos\left(kx-\omega t+{\varphi \over 2}\right) $$

That is, summing two sinusoidal waves identical except in phase together, the result is a sinusoid of amplitude proportional to $\cos(\varphi/2)$.

A quick graph of that:

enter image description here

Keep in mind we aren't looking at the resulting wave as a function of time, we're looking at the amplitude of the resulting wave as a function of phase difference between the two component waves. That they are both sinusoids is coincidental.

Antenna patterns do not usually distinguish polarity, but sometimes you find one that does and you will see each lobe alternates in polarity. What's happening is for the purposes of the radiation pattern we really just care about the absolute value of the amplitude of the resulting wave. So now, a graph of $|\cos(\varphi/2)|$:

enter image description here

Now one sense of "sharp" means an abrupt change in direction. > is sharp. ) is blunt. You could say something is sharp if there is a discontinuity in the derivative. The nulls certainly look like they do, but if you consider the polarity of the lobes, there is not actually a discontinuity. That's just an artifact of taking the absolute value of the amplitude for the purposes of the polar plot.

Another sense of "sharp" is "not wide". The reason it's hard to make very narrow lobes is mathematically similar to the reason a pulse train has infinitely many harmonics. Our simple example considered only 2 waves from 2 antenna elements, but if you want more complex antenna patterns you can create them by adding together spherical harmonics just as you can create any periodic waveform by adding together harmonically related sinusoids. However, the number of antenna elements, and thus the number of spherical harmonics that can be added together, is limited by practicality.

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  • $\begingroup$ I think that’s it. Antenna far field pattern gain is often or usually defined or plotted in terms of log magnitude, which is a non-linear function with discontinuous derivatives at zero. $\endgroup$
    – hotpaw2
    Jul 21 at 21:03
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Well, antenna gain plots are logarithmic, and $\log(\sin^2(x))$ hits negative infinity for every zero transition of the sine's power, so while the power of some directivity pattern may actually touch its minima as gently as its maxima (it depends on just what kind of transition is there, though), in a logarithmic gain plot you get to see a lot more drama.

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  • $\begingroup$ It might be good to augment your statement with an example. Even if it were about as easy to construct an antenna where a certain amount of rotation from either a node or a null would yield a five percentage point change in signal strength, recognizing the difference between a signal that's 100% of maximum versus 95% is much harder than recognizing the difference between a signal that's 5% of maximum and 0% of maximum. $\endgroup$
    – supercat
    Jul 21 at 17:45
  • $\begingroup$ Hello and welcome to ham.stackexchange.com! $\endgroup$
    – rclocher3
    Jul 22 at 1:28
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As @glen_geek points out in his answer, "getting to zero" is pretty definitive.

If you'll accept "very narrow lobes" as an acceptable interpretation of "sharp lobes" then I'd add:

Antenna systems can achieve a very narrow beam-width with some of the following techniques:

  • an array of elements as in a yagi-uda
  • an array of arrays as in a stacked yagi config, or even in a grid config
  • a parabolic antenna
  • a field (farm) of powered vertical elements, each phase controlled to manage maximum constructive addition in a particular direction (and has the advantage of being electronically steerable when computer controlled)

Not all of these techniques are feasible or scalable at all wavelengths, but that's another question.

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A null is a state where parts of a system are in nearly perfect balance. A lobe represents a state where they are maximally unbalanced. If a system is balanced, even a small change can make it quite obviously imbalanced. By contrast, if a system is maximally unbalanced, a much larger change would be required to make the system be noticeably less unbalanced.

A directional antenna, for example, will often have two symmetrical or mirror-image components which radiate or receive most strongly in different directions, but operate out of phase. At a direction halfway between their peak directions, the effects of the two components will cancel out almost perfectly. Changing direction will cause one element to become slightly more effective while the other becomes less effective, thus creating an imbalance which is much more noticeable than the change in signal strength from either side would be in the absence of the other.

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  • $\begingroup$ Hello and welcome to ham.stackexchange.com! $\endgroup$
    – rclocher3
    Jul 22 at 1:29
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    $\begingroup$ @rclocher3: Do you like the explanation? It's simpler than the others, but I think it hits a point the others miss: the fact that the closer something is to being balanced, the easier it is to notice changes. $\endgroup$
    – supercat
    Jul 22 at 14:07
  • $\begingroup$ I like it so far @supercat, but would you please edit your question to better explain what you mean by "balance"? Some pictures or sketches might help! $\endgroup$
    – rclocher3
    Jul 22 at 14:20
  • $\begingroup$ @rclocher3: I'm not an artist, but I added a description of what "balance" would mean in this context. Does that help. $\endgroup$
    – supercat
    Jul 22 at 15:40
  • $\begingroup$ Well I'd still prefer a graphical explanation, but I can't imagine how to illustrate the idea either ;) The second paragraph does help, so +1 to you! As you know these comments don't help future readers, and they do clutter the question, so I'll delete mine in a day or two, and I suggest you do the same. Again, welcome to the group! I'm sure your computing and electronics experience will enrich our discussions. $\endgroup$
    – rclocher3
    Jul 23 at 0:35

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