You don't seem to have taken account of the rotation of J between the elements, as you only have Jo which does not vary with n. If J rotates with element position you need take account of this using a rotation matrix or something equivalent.

Are you able to reproduce here the equations you are comparing with?

Malcolm White the rotation concerns only the spatial distribution of the sources (that is, what I‘ve written in cilindrical coordinates), not also the polarization. All the sources have the same polarization. This is the reason why no rotation matrix are needed.

The theoretical formulation of the radiation pattern is not complicated attached a chapter which clearly describes the radiation pattern of several structures including circular antenna array, just here ! attention to the level of spacing between antennas to avoid the electromagnetic coupling also it is necessary to choose the efficient method for the synthesis of radiation diagram and the phase and amplitude controls.

Ridha Ghayoula Unfortunately I cannot find a proof in the pdf you sent me (even if it is in French, so I may have missed something). What I would like to understand is if there is a specific error in the math passages I wrote in my pdf.

Anyway, if I repeat the analysis thinking the whole problem 'in reception' rather than 'in transmission', the formula I get is exactly like the one reported in the book. So I would say that I am almost certainly doing something wrong when I set the problem in transmission (ie with the vector potential, as I did in the pdf), because the results of both approaches must be identical, for reciprocity.

Malcolm White The equation I am comparing with is attached here.

A trivial correction - the angle should be 3 delta phi in the figure.

I can see no terms in your equation that refer to the rotation of each antenna element, nor for the summation of terms for the parts that make up each element. You need a summation of terms like EL() for each element, inside the summation of elements. You have stated that there is a current distribution around each element of the array. This needs to be integrated first, then summed over all the elements.

Malcolm White If 've understood correctly what you are telling to me, the sum of EL() you are referring to, is exactly the last equation in my pdf.

The rotation of the elements is encoded in the equality that I write when I say ' Then, using the property J(n)(r′,φ′,z′) =a_n * J(0)(r′,φ′−n∆φ,z′) ...'.

J(n)(r′,φ′,z′) =a_n * J(0)(r′,φ′−n∆φ,z′) just seems to be the pattern of one element J(0) multiplied by a_n which is a complex number you have not specified, adjusted for phase because of its position in the array. What is missing may be included in a_n, which is a complicated function due to the rotation of the distribution around each element centre but not its polarization, which needs to be calculated separately for each element. I may be wrong in my understanding, but I only see the summation of n identical elements, not the summation of n elements each of which is a different result of a summation over the element distribution.

Each element will have a different polar diagram, unless the distribution around the element has circular or n-fold symmetry, because the polarization does not rotate with the distribution function. The contributions from all these different polar diagrams then need to be added together in one particular direction taking account of the phase difference due to the position of the centre of each element.

Malcolm White

----------J(n)(r′,φ′,z′) =a_n * J(0)(r′,φ′−n∆φ,z′) just seems to be the pattern of one element J(0) multiplied by a_n which is a complex number you have not specified, adjusted for phase because of its position in the array.----------

J(n)(r′,φ′,z′) is not the pattern of the element n, but it is the current distribution of the n-th element; the pattern of the n-th element is the usual integral of this distribution.

----------What is missing may be included in a_n, which is a complicated function due to the rotation of the distribution around each element centre but not its polarization, which needs to be calculated separately for each element.----------

a_n are the coefficient used for beam-shaping and beam-scanning, they can be considered as known constants in this analysis, they are not relevant.

----------I may be wrong in my understanding, but I only see the summation of n identical elements, not the summation of n elements each of which is a different result of a summation over the element distribution.----------

The elements are not identical, because there is a rotation between them, and this fact is written in the equation. Then, the sum is done considering this difference, because as we can see from my last formula in the pdf, the element pattern depends on 'n', and so it doesn't come out from the summation sign.

----------Each element will have a different polar diagram, unless the distribution around the element has circular or n-fold symmetry, because the polarization does not rotate with the distribution function. The contributions from all these different polar diagrams then need to be added together in one particular direction taking account of the phase difference due to the position of the centre of each element. ----------

Exactly, and this is what I do.

Trying to make each single mathematical step better understood, I added an identical treatment, which uses the same principle, for arrays whose elements are only translated, and not rotated. I am attaching it here as a new pdf. As you can see, the final formula is the known one, which is correct, because it shows how 'element factor' and 'array factor' splits.

The analysis done for the circular array follows the same identical steps, except that when I insert the rotation in the formula, an exponential term independent of the integration variables no longer appears, as it did in rigid translations.

Thank you - I'll have another go.

I solved! The fact is that in my analysis the phase center of all elements of the array is the origin, the center of the circle on which they are arranged. So, it's as if I had to compare my formula to his, using R_a = 0 in his, and then they match. By making a change of variable in my last integral, bringing the phase center to a point within the current distribution, we get the same formula as in the book. So, in summary, there was no mistake, but it was the definition of 'EL' that was different.

Thank you both for the time you have dedicated to me Malcolm White and Ridha Ghayoula , you have been very kind.