My english is limited.have i explained my puzzle clearly'？

No one?

As distance between antenna elements increase , grating lobe formation starts and this results in reduction in gain...

Shrikanth Reddy : The paradox is that in the first poiont of view, the gain changes with the distance, however in the second point of view, the gain do not change with the distand.

The power level of a signal is affected as it propagates, which is more apparent when the distance relates.

The antenna gain is not affected.

Ma Han- Qing: your first point of view is absolutely correct, because as element distance varies the nature of array factor pattern will chance. Gain of array antenna will be decided by the pattern multiplication of AF and antenna pattern. Apart from this as i told earlier that as distance between antenna elements increase , grating lobe formation starts and this results in reduction in gain. these grating lobe will derive power from main beam and thus reduce the gain.

Shrikanth Reddy : yes, i also known the first point of view is correct, but i do not known what is wrong with the second view. that is way i call it as a paradox .

Ma Han-Qing : In your question, it is given that elements are isolated, now we take 'isolation' as less mutual coupling between the elements. Now out put of individual elements will add up and multiplied by array factor to give final out put. Now if you see the equation of Array factor for an array it includes distance between the element. So if AF varies with distance so does the antenna pattern and consequently gain have to change with distance. Therefore statement "gain do not change with distance (elemental distance) " seems to be in correct.

I do not see it as a paradox. Increasing the inter-element spacing results into a larger antenna aperture, therefore, into a higher gain in *certain directions*. Increasing the aperture, *exactly the same amount of energy* is better focused towards certain directions, so the gain is larger just *in certain directions*, and smaller in other directions in the vicinity of the main lobes, that are now narrower. Integrated over all angles, the radiated energy is exactly the same.

To Traian:

you explain the first point of view very well. however you do not defeat the second point of view at all. a question called as a paradox is just a question Both side of it is right . try to defeat the second point of view!

To Shrikanth Reddy:

you do not defeat the second point of view too. remeber that ,please, the first view is right do not means the second view is wrong. try to prove the second of view is wrong directly.

It seems no one understand me.

My quesiton is it seems both views are right. it is also well kown that the first view is right. however, the second is difficult to defeat. you can not say:the first is right so the second is wrong. that is unfair.

It is because, if I understood correctly, I find the second statement incorrect: "In this point of view, the gain of the array has nothing to do with the element distance". Yes, it does. Depending on the spacing, the radiated electro-magnetic fields corresponding to each element will undergo different constructive/destructive interference, and this is exactly why you get a different radiation pattern. The independent radiated EM waves will sum up with different phases. You are saying the spacing does not matter because the elements are being isolated from each other. This is the misleading point. In an ideal scenario, although the elements would be perfectly isolated from each other, the waves still DO interfere in the medium, both in near and far field. Here, I assume that by "isolated from each other" you mean "there is no mutual coupling in the near field". The total energy is preserved, it is just distributed differently. I reiterate my previous statement, and even strengthen it. I do not see any paradox at all, and no "two sides of the story". I hope I understood well your question.

I agree with Traian Abrudan, No mutual coupling does not mean total out put energy will remain invariable, as Traian Abrudan stated about const/distructive addition of patterns in far field, this const/distructive addition is decided by your elemental distance (as in case of AF), so if patten varies with element distance, so does the gain (for certain direction).

To Traian: I am sorry to say that you still not understand me. it may because that my english is limited. you keep to say the first view (gain change with distance)is right, but have on comment on the second view. try to defeat this :"However, as the elements have been assumed to be isolated from each other, the output energy of each element and the total array are invariable. The maximum field, which is the vector sum of the field of each element, is also invariable."

defeat this process but not to defeat the conclusion, please.

we known that Gain defined as : E(in the maxium direciton)/total radiation power. now the E is sum(Ei) i=1,2...N. as the element is isolated. As the elements are isoluted and in phase. sum(Ei)=N*E wich is constant. and the power is constant (as you word: The total energy is preserved) too.

I also know the first view is right. as my word: "This is true for real antenna arrays. "

how ever , i do not know how to defeat the second view too.

by the way. i am not agree with Shrikanth. " No mutual coupling does not mean total out put energy will remain invariable" . Try to image we put one antenna with radiation power P in to a line, and then , the second one at any distance in the line but in phase with the first one (as no mutual coupling, power will be 2P), the third one(will be 3P) , the 4th one (will 4P). as no mutual coupling, the antenna will not effect each other. if you say the radiation power will be changed, so where are the superfluous power going to? that is way i think "isoluted point source is impossible".

Here : if u put elements of array at certain distance.. This inter elemental spacing will decide

1)whether the antennas will be in phase or out of phase ( even if feed in phase)

2) the nature of energy summations i.e. The direction where the power is going to or not going to (constructively or destructively).....

Now say if antennas elements in array are very far away, their behavior will still be govern by their inter elemental spacing ..As the statement says "The maximum field, which is the vector sum of the field of each element, is also invariable." the field consideration here should be with respect to direction and elemental spacing ...because total field of an array is not just a vector sum but also multiplied by array factor...

I already answered your second concern, i.e., that although there is no mutual coupling between elements, the waves WILL interfere in the medium and create a different antenna radiation pattern, which translates into different gain in different directions. You formula sum(Ei)=N*E assumes that the waves always add constructively in order to produce the same gain in direction of main lobe, which is completely FALSE. Do a simulation and add the waves with whatever phases you like, in order to see the misconception you are subject to. The width of the narrowest main lobe you can get by perfect phase matching will decrease as the antenna aperture increases. The larger the antenna (hence the element spacing), the larger amount of energy the antenna collects from certain directions (works also the other way, i.e., radiates in certain directions). It is the basic concept of aperture. Here is a simple analogy: it is not the same if you have a lens (or mirror) of diameter D, or a lens of diameter 10*D. The larger lens collects 100 times more light than the small one. It is about the energy flux that passes through a surface. Basic Physics. Same happens with the antenna gain. I still do not understand your puzzle.

Array directivity increases monotonically with element distance only when there is a single main beam (no grating lobes). When grating lobes appear, directivity suddenly decreases and then increses with distance again until the appereance of the next grating lobe. So, array directivity oscillates with increasing distance. But the oscillation amplitude decreases and, therefore, directivity asymptotically tends to a constant when the element distance tends to infinity. The value of that constant is the square of the sum of array coeffcients over the sum of the squares of that coeffcients.

OK, I try to explain my puzzle more clearly.

For a give omnidirectional element with radiation resistance Z and radiation power P, at a distance R faraway the antenna , we have its Radiation far field: E=k*sqrt(P/Z)/R, where k is a certain coefficient. This comes from E=KI, means the far field E is always proportional to the exciting current I. If we image we have an omnidirecitional antenna element with efficiency 100% linked to a power source with a FIXED power output P, its far field will always be E=k*sqrt(P/Z)/R. Then, we add another element which is also linked to a FIXED power source, Its far field is still be E=k*sqrt(P/Z)/R.

if we put all N elements in line, and with the same distance D and in phase. In the broad side its far field will certainly be E_broadside=sum(Ei), that is E_ broadside=Nk*sqrt(P/Z)/R. as all the energy is radiated into the space, the radiation power must be N*P. Thus, the gain defined as (2*pi*R*R*E*E/radiation Power) will be G= (k*sqrt(P/Z). It SEEMS fixed. And this comes from that we image that the element is linked to a FIXED power source. This seems impossible! Because we have another way to get the E_broadside, that is, as you say, we must fist calculate the radiation pattern integral, get the pattern Gain and then, E_broadside=sqrt(NP/2*pi*R*R)*Gain. And the Gain is a function of element distance D.

and we will find E_broadside will not be equal to Nk*sqrt(P/Z)/R. Thus, we will find one thing must be changed, that is the radiation resistance Z! Means that, the radiation resistance of a lonely element must be different from that of the element in an array. However, this is conflict to “no Mutual coupling”. That is my puzzle.

As you said, in an antenna array, "one thing must be changed, that is the radiation resistance Z! ", even you wish the elements were totally isolated. The mutual coupling between the elements CAN NOT be assumed to be eliminated if you still consider the far-field interference. For an isolated antenna in free space, the input impedance is mainly determined by the interaction between the accelerating electrons and radiating fields around them; for an antenna array, the radiating fields are contributed by all the elements. This mutual coupling exists for every antenna array.

The directivity (or the gain if the efficiency is equal to 1) at broadside is given by the power at broadside per UNIT SOLID ANGLE divided by the total radiated power. The unit solid angle part is important. The total power and the peak electric field are constant, but if the beamwidth shrinks, the power per unit angle at the peak will increase. I hope this clarifies it.

Minor correction:, the directivity is the power at broadside per unit solid angle divided by the total radiated power and multiplied by 4pi (because it has to be normalized to the directivity of the uniform radiator)

I think you should point out that by locating elements on aperture of any array, we actually sample the incoming signal at different sample intervals (spacing), and thus spacing plays a role in construction of the pattern (which may be assumed as a metric of reception or transmission quality of array). If you increase the spacing, you are increasing the sampling interval, and thus, received signals are more likely to interfere, which from pattern point of view may be inferred as the related change the sidelobe topology, especially appearance of grating lobes (one cannot distinguish between actual and image or misleading source). Periodicity in regular spacing is a vital issue. First take a Tchebysheff array, then increase the spacing. You will see, at periodic instances, the grating lobe are appearing, and no change in sidelobe topology. And yes, the total power content will change with spacing, because you are changing the nature of pattern and thus the pattern integral will include nature of pattern. If the pattern is consisting grating lobes, it will be inferred as multiple main beams. Hope this will help you.

now , i have understood this..

omnidirectional elements do have mutual coupling.  seeing the book "modern antenna design" by THOMAS A.MILLIGAN.  the coupling changers the radiation Z and thus the max field. And ,with coupling. we can caculate the antenna gain without doing integration.

Here's my interpretation:

The implementation of an antenna array is nothing more than a discrete sampling of a particular current (or voltage) distribution in free space. Knowing this, increasing the amount of elements in an array (without changing the inter-element spacing)  is to sample a larger distribution. As you know, larger aperture = higher gain. However, if we were to increase the inter-element spacing but not the number of elements, the sampling of the excitation distribution becomes less and less accurate and the phase difference and/or polarization between the signals from each element at any observation point in the far field changes, leading to pattern breakdown.

For instance, consider a two-element array of collinear hertzian dipoles for simplicity. In general, we think of the E-field amplitude at broadside (theta = 90 degrees if the elements are aligned along the z-axis) to be the sum of the E-fields from each element, but this is only the case if the distance between the elements is small enough for the paraxial approximation to be valid. If we were to increase the inter-element spacing indefinitely, the fields from each element would no longer be co-polarized at theta = 90 degrees, complicating the field calculation and leading to unexpected interference affects.

In short, simply increasing the spacing causes our assumptions to break down and a paradox only arises if the assumptions are said to hold in all cases, which is not true.

https://pdfs.semanticscholar.org/d2e5/71690a30ed83ebf0651dbcd6f81f32cdf91f.pdf