The paper introduces a method for measuring angular bearing errors in an antenna-radome system in a two-dimensional angular region of scanning of electronically steerable antenna, i.e. phased array antenna, active phased array antenna. The measurements were based on an antenna measuring collimator system - “compact testing ground”. The mathematical expressions used in processing the obtained data are given. To make a matrix of angular bearing errors, measurements are carried out at different angles of the heel of the antenna - radome system when the phased array antenna beam is deflected in oblique planes. To perfect the method at the initial stage, we used a quick-release model of the radome, which has the ability to introduce angular bearing errors

Представлен способ измерения угловых ошибок пеленга в системе антенна - обтекатель в двумерной угловой области сканирования антенны (фазированной антенной решетки, активной фазированной антенной решетки) с электронным управлением лучом. Измерения проводились на базе антенного измерительного коллиматорного комплекса («компактный полигон»). Приведены математические выражения, используемые при обработке полученных данных. Для создания матрицы угловых ошибок пеленга измерения осуществляются на различных углах крена системы антенна - обтекатель при отклонении луча фазированной антенной решетки в косых плоскостях. Для отработки методики на начальном этапе использовалась быстросъемная модель обтекателя, обладающая способностью вносить угловые ошибки пеленга

Errors in determining antenna boresight position (angular bearing errors, ABE) occurring in the presence of a radioparent radome (RPR) [

Given in Fig. 2 is a PAA (pos. 1) whose aperture centre (pos. 2) is associated at the same time with the spherical (θ, Ψ) and Cartesian (X, Y, Z ) coordinate systems. Before installation of the RPR (pos. 10), antenna rotary support device (ARSD) is set to an arbitrary heel angle Ψj– (pos. 5). Under arbitrarily selected spatial angle θi– (pos. 7) of deflection from the normal to the antenna aperture (pos. 8), controlling angles Lx and Ly (pos. 3 and 4) between respective axes X, Y and beam setting direction (pos. 6) are calculated by the formulas

cos Lx = sin θi cos Ψj; (1)

cos Ly = sin θi sin Ψj, (2)

and the array phasing angles (αх; αy) transmitted from ABR computer to the PAA beam steering unit, by the formulas

αх = 90 - Lx; (3)

αу = 90 - Ly. (4)

Given such calculation of phasing angles αх, αу and with system heel angle Ψj being constant, any angle θi of beam deflection from the normal, selected from those possible for a given PAA, will be setting the beam into a plane (pos. 12) coinciding with the ARSD azimuthal turn plane (pos. 14). If in so doing the ARSD azimuthal platform with PAA being tested is turned to angle (−θi), then the normal to the electromagnetic wave plane front incident on the PAA aperture in the far zone (in this case, collimator operating area) coincides with the beam setting direction. If the PAA being tested has outputs of both difference channels (Az and El), by slightly turning ARSD about its azimuth axis (pos. 11) in the area of angle (−θi), the coordinates of the cut minima of both difference patterns can be changed simultaneously.

It is known that the lines of zero levels of the spatial RP of planar-PAA difference channels are designated in the coordinate system of guiding (steering) cosines (u, v) by straight lines along axes u or v. Given in Fig. 3 are spatial RP of the difference channels.

The cartographic lines of pattern cuts in Fig. 4 schematically represent spatial difference RP (Az – green cut lines, El – red ones) in the coordinate system of guiding (steering) cosines U = cos Lx, V = cos Ly. Lx и Ly – earlier determined controlling angles for a PAA set to heel angle Ψ. Then, in the coordinate system of guiding cosines, a possible area of PAA beam adjustment will be restricted by a unit circle, and beam setting coordinates on this plane will be fully determined by the angle of PAA beam deflection from normal to aperture θo and the angle of PAA setting by heel Ψ. Two characteristic points can be singled out on plane (U, V).

In point 1, before RPR installation, a PAA beam is set, according to formulas (1)–(4), as per spherical coordinates θ0, Ψ and corresponds to the precise boresight direction towards the electromagnetic wave plane front arrival from “target” (in this case, from collimator of the measuring complex). On the plane of the steering cosines, this is a point with coordinates (u0, V0). It is in point 1 that intersection of zero level lines of the spatial difference RP, shaped by the PAA being tested, occurs. It means coincidence of the coordinates of the minima of difference RP cuts simultaneously measured in the Az and El channels (Fig. 5, а), under which θα = θу, coinciding with beam deflection angle θ0.

In point 2, after RPR installation, the PAA beam is still set as per spherical coordinates θ0, Ψ to the direction of the electromagnetic wave plane front arrival from target. However, because of refraction in the RPR, now the direction of electromagnetic wave plane front arrival corresponds to the initially set boresight direction no more. In this way, point 2 (as one of the possible points) with coordinates (U1, V1) corresponds to the changed boresight direction caused by RPR installation. Intersection of zero level lines of the spatial difference RP, shaped by the PAA – RPR system being tested, occurs in point 2. Since the PAA – RPR system is still phased as per coordinates θ0, Ψ the actually measured minima of the cuts of both spatial difference RP, determined by the end of vector sin θα and sinθу, will be fixed at angles θα and θу, respectively. In that case, the fixed angles of the measured minima θα and θу are not equal and do not coincide with θo (Fig. 5, b).

On the plane of the guiding cosines, the coordinates of point 1 are U0 = cos Lx, V0 = cos Ly, and in variables of the spherical coordinate system they are expressed by the above formulas (1) and (2). Applying the method of ARSD mechanical azimuthal scanning in the vicinity of point (-θο) and measuring angles θα and θу (the initial setting angle θo and Ψ are known), which changed after RPR installation, and also taking into account that

u1 = u0 + du; (5)

du = (sin θ0 - sin θα )cos Ψ; (6)

V1 = v0 + dv; (7)

dv = (sin θу - sin θ0 )sin Ψ, (8)

we calculate du, dv, by formulas (6), (8) and u1, v1. by formulas (5), (7).

Considering that

Lхo = arccos u0; (9)

Lx1 = arccos u1; (10)

Lyo = arccos v0; (11)

Ly1 = arccos V1, (12)

from equations (3) and (4) we define the increment of component Δαх to the initial phasing angle of the PAA being tested by coordinate αх, caused by RPR installation on the PAA, as

Δαх = Lx1 - Lхo. (13)

Whereas the increment of component Δαy to the initial phasing angle of the PAA being tested by coordinate αy , caused by RPR installation on the PAA, we define as

Δαy = Ly1 - Ly0. (14)

In fact, it is Δαх and Δαy that are the sought boresight errors for a single element of the ABE matrix, calculated for the position of PAA beam setting with spatial spherical coordinates (θ0, Ψ) which corresponds to the array phasing coordinates αх and αy . These data are transmitted from ABR computer to the PAA beam steering unit with each clock cycle of beam readjustment.

Making the necessary substitutions and further transformations with (1)–(14), we can obtain a more general representation of the expressions for Δαх, Δαy, as a function of arbitrary (permissible for the PAA being tested) spherical coordinates of system installation θ0ί, Ψj.

In this case, boresight error components Δαxi, j, j, Δαyi, j proper, introduced by the RPR at angles θ0 = θ0ί, Ψ = Ψj of the antenna – radome system installation in the variables of spherical coordinate system of the antenna can be calculated by the formulas

where Δαxi, j – component of the boresight error introduced by radome by angle αх (at θ = θοί, Ψ = Ψ);

θαί – actual measured angle of the minimum of difference spatial azimuthal pattern cut (at θ = θ0ί, Ψ = Ψj) after radome installation;

Ψj – current value of antenna system heel angle relative to the ARSD azimuthal scanning plane;

θoi – current value of angle θo in PAA spherical coordinate system between the normal to PAA aperture and direction to the minimum, formed by spatial difference RP, before radome installation, under which the measured minimums in the cuts of difference RP coincide;

Δαyi,j – component of the boresight error introduced by radome by angle αу (at θ = θoi; Ψ = Ψj);

θуi – actual measured angle of the minimum of difference spatial elevation pattern cut (at θ = θoί; Ψ = Ψj) after radome installation.

It should be stressed once again that it refers to calculated errors in the coordinate system of PAA phasing angles (αх, αy). Although the obtained errors can be recalculated to other coordinate systems as well, e. g., those of the carrier, for their subsequent compensation in the ABR boresight channels this type of their expression seems the most appropriate.

Supposing that the initial PAA installation on ARSD is such that there is a coincidence of beam setting planes (pos. 12, see Fig. 2) of ARSD azimuthal turn plane (pos. 14) with the horizontal plane, then intersection of the minimum-level lines of the difference spatial patterns formed by PAA occurs in the points (pos. 13) (i. e., condition θαί = θyi is satisfied). ). In the measuring complex display screenshot (see Fig 5, а) it appears as coincidence of the coordinates of the measured cut minima of both difference RP. After RPR installation on the antenna, new changed coordinates (Fig. 5, b) of the cut minima of the difference RP θαί and θуi. are measured. In this case the proposed mathematical model is functioning, and the boresight error components Δαxi,j и Δαyi,j can . be determined by formulas (15)–(16) from the magnitudes and signs of θai and θуi deviations from the initial θoi, where condition θαί = θyi, was satisfied. It means that they can be calculated in the points (pos. 13). In so doing, the section lines (pos. 15) of the RPR surface and PAA beam setting plane (pos. 12) on which those points lie will determine one of the lines of the ABE general matrix.

Subsequently, similar measurements can be made for the entire RPR surface, i. e., under all possible discrete settings of the PAA – RPR system by heel angles Ψj and beam deflection angles θoi. In this way, a set of lines, and, accordingly, a matrix of ABE components for an RPR of arbitrary shape, can be obtained. Thus, for instance, in the case being considered (see Fig. 2), RPR axis (pos. 9) does not coincide with the normal to PAA aperture (pos. 7). Shown in Fig. 6 is a complex surface of an arbitrary RPR occurring within the PAA beam scanning area. The section lines of its surface and the points on them by which ABE components (Δαxi,j and Δαyi,j) are measured are shown in different colours (see Fig. 6).

It was said above that for further correct measurements of ABE introduced by RPR, before installation of the latter it is necessary to take precise bearing of the PAA being tested to the plane front of incident electromagnetic wave. Only in this case the measured minima of the spatial difference RP cuts coincide, and then it is possible to make calculations according to the proposed method. Moreover, this condition shall be satisfied in any point available for measurement as per the given method, i. e., under any possible θ0 = θoi, Ψ = Ψj.

Fig. 7 provides a schematic representation of an ARSD similar to that used in the measurements. Two elementary cases are presented, where, due to the errors of PAA initial setting by heel (∆ψ) or by deflection angle (∆θ), the azimuthal turn plane does not coincide with the true horizon plane.

In the most general case, the real plane of ARSD azimuthal turn in Fig. 7, together with the PAA rigidly connected to it, may take even more complex arbitrary spatial position. Herewith, angular coordinates αxi,j and αyi,j, calculated by formulas (3), (4) and determining array phasing, fail to set the beam strictly along the boresight direction towards the plane front of electromagnetic wave arriving from the target (collimator). In such points, condition θα = θy is not satisfied, which means that should ABE be subsequently calculated relative to those coordinates, the error will be unacceptable.

In certain exceptional cases the real azimuthal turn plane can be reduced to the conventional plane of the “true” horizon through an initial mechanical adjustment of ARSD, together with the PAA installed on it, in one of the points of system installation. It can be achieved, for example, by adjusting the ARSD mount supports (pos. 2 in Fig. 7). However, the experience of practical measurements, as well as their enormous potential volume, goes to show that this is a blind alley. To obtain a correct initial setting under any possible θo = θoί, Ψ = Ψj is physically infeasible. It is exactly for this reason that for practical implementation of the proposed ABE measurement method the latter is based on the principle of electronic compensation of the initial PAA beam setting when measuring ABE (before RPR installation).

For compensation of the errors of initial PAA beam setting by boresight direction towards the plane front of incident electromagnetic wave, a computation method is proposed. Its essence consists in primary compensation of beam setting error of a PAA phased as per coordinates θo = θοί, Ψ = Ψj by means of PAA itself. For that purpose, same as in the described ABE measurement method proper, it is proposed to measure the initial divergence of θa and θy and calculate additives ∆αxi, j комп и ∆αyi, j комп by formulas (15), (16), the only difference being that, taken with the respective sign and added to the initial phasing coordinates αxi, j нач и αyi, j нач (at beam setting angles θο = θoi, Ψ = Ψj), they will act as the compensating ones and are supposed to bring the system to a point where condition θα = θy is already satisfied. If such compensation procedure is performed for all points of the array intended for subsequent ABE measurement, it can be said that after installing RPR and measuring ABE in the points where compensation has been performed, the calculated Δαxi j and Δαyi j are indeed the boresight errors introduced immediately by RPR. For this reason, this method for determining ABE components will be hereinafter referred to as the compensation method. Given in Fig. 8 are the data of practical measurements of θa and θy (before RPR installation) by the signals of two difference channels under the initial setting of the tested PAA by heel Ψ = 60° and beam deflection from normal θo = 45° . Fig. 8 (а) shows divergence of the coordinates of zeros of the measured difference patterns, and Fig. 8 (b) illustrates the result of the compensation performed.

For all subsequent ABE measurements (after RPR installation), a point taken as the initial setting point should be a new one, with account of the compensating additive ∆αxi, j комп and ∆αyi, j комп.

Measuring complex AKK1-12. Equipment of the complex. Automation of measurements using Stend BEK software

Given in Fig. 9 is a block diagram of antennacollimator complex (AKK1-12), on the base of which the measurements were performed.

Large arrays of data, measured and processed according to the proposed method, required development of a special-purpose program. Software (SW) product Stend BEК was developed and tested on the complex. Stend BEK SW interface is shown in Fig. 10.

In addition to the usual RP measurement, when three channels of the tested PAA (sum channel, azimuth and elevation difference channels) are used simultaneously, it makes it possible, using the proposed method, to calculate ABE occurring in the PAA – RPR system. At the initial stage, before RPR installation, the minima of the cuts of both difference RP are measured simultaneously on each one of the selected cuts by heel angle Ψj and by the generated job file for θoi , and, from the coordinates of the initial divergence θa and θу,

compensating additives in each point of PAA initial phasing (∆αхi, j комп and ∆αуi, j ) комп from the job file are computed. Then the measurements are repeated for the same points with RPR installed. Next, considering the found compensating additives in the coordinate system of phasing angles αх,j and αу,j , ABE components (Δαχι·,j и ΔαyiJ), introduced immediately by the RPR itself, are calculated. Proceeding from these measurements, a complete ABE matrix is subsequently generated. Besides, since in the course of measurements and ABE calculation a sum RP (before and after RPR installation) is measured too, it is not at all difficult to also compute RPR transmission factor (TF) in the same points where the ABE measurement was taken. So, even though such task was not set, based on those data, a complete TF matrix for the entire RPR surface can be easily generated.

Before proceeding to discuss practical measurements, it should be appropriate to consider the occurring errors, inherent in both the measuring complex on which the measurements were made and the proposed method of conducting them. They can be divided into systematic and random ones.

Systematic errors and possibilities of their compensation. For the most part, an error compensated by the method described above can be categorised as a random error, i. e. such that is determined by mechanical misalignment between the movement plane of tested PAA and a hypothetical plane of measurements (“true” horizon plane). However, even if the initial compensation was performed correctly in all the points intended for measurement (i. e., for all possible θoi, Ψj) in the absence of RPR on the PAA being tested, there is another risk arising after its subsequent installation. If scanning planes of the tested PAA with and without RPR coincide completely, i. e. installation of extra weight on ARSD does not change spatial position of PAA with RPR relative to PAA positions before radome installation, for all possible θoi, Ψj, then no additional error occurs. Yet, practical measurements demonstrate that this is not the case. Installing on the tested PAA even a fairly light (about 12 kg) RPR simulator mockup, which is described below, results in its spatial position offset relative to the initial position. In such an event, system non-returning to the initial reference datum point will lead to addition to the measured real ABE components, associated solely with wavefront refraction by the radome, of unaccounted components of both errors caused by the changing spatial position of the system. After that it is impossible to discern contributions from one and the other factor to the values of the measured ABE components. Fig. 11 illustrates available measures taken to prevent errors introduced by the ARSD mechanical system (mainly in the elevation channel drive).

A laser pointer sensor, secured by means of a rigid bracket before installation of RPR on the tested PAA, produces a light spot on the collimator surface (we consider the collimator a stationary object). The crosshairs of theodolite optical tube are laid on it. Azimuth and elevation of the laser pointer spot (and, accordingly, of the tested PAA itself) are taken as per readout scale of the instrument in the coordinate system of the theodolite. This procedure is performed for all heel angles Ψj, at which ABE measurement is intended. The azimuthal turn angle at which such ARSD referencing occurs was in this case the same – θoi = 0 (more exactly, the azimuthal angle of the found electrical axis at heel angle Ψj). After RPR installation, the laser pointer spot becomes offset due to mechanical sagging of the ARSD elevation drive (possibly because of mechanical backlash) together with the PAA installed on it. To bring the PAA – RPR system to the datum point by means of ARSD electromechanical elevation drive according to the theodolite coordinate readout scale, the system is returned to the initial reference point. This referencing procedure (as per a table prepared before RPR installation) was performed for all possible heel angles of the system – Ψj. For the worst case the accuracy of compensation with the use of ARSD electromechanical drive in the theodolite coordinate system was 3 angular minutes. It is shown in Fig. 11 that, with account of mutual positioning of ARSD, the collimator mirror, the tested PAA, and the theodolite, when translated to the antenna coordinate system, the accuracy will be of the order of 4.2 angular minutes.

Unfortunately, compensation of elevation drive mechanical “sagging” per a stationary reference object (collimator) was only performed at zero angles of ARSD azimuthal turn, and it is hard to say what will be the maximum error of ABE components measurement associated with misalignment of planes under other angles of ARSD azimuthal turn θoi. Control of object (PAA – RPR) positions throughout the entire range of setting angles θoi, Ψj is possible, although it would require a substantially greater quantity of additional position sensors. In the considered case there were neither time nor material resources for further, more profound study of the matter.

Random errors. Attributed to this category can be, first of all, mean-square phase error of an array with a given quantity of phasecontrolled elements in the aperture and a known number of binary bits of phase shifter control. For that reason, within the framework of the proposed method, both the initial compensation error and the error with which subsequent calculation of components (Δαxi, j Δαyi, j) was performed are random as well. After the measurements, calculation of the ABE components introduced by RPR is done by formulas (15), (16), which in certain cases can be simplified to the expressions:

where Δαxi,j – component of the boresight error introduced by radome by angle αх (at θ = θoi; Ψ = Ψj);

Ψj – current value of antenna system heel angle relative to the ARSD azimuthal scanning plane;

θoi – current value of angle θo in PAA spherical coordinate system between the normal to PAA aperture and direction to the minimum, formed by spatial difference RP, before radome installation, under which the measured minimums in the cuts of difference RP coincide;

∆αуi, j – component of the boresight error introduced by radome by angle αу (at θ = θοί; Ψ = Ψj);

θai – actual measured angle of the minimum of difference spatial azimuthal pattern cut (at θ = θοί; Ψ = Ψj) after radome installation;

θyi – actual measured angle of the minimum of difference spatial elevation pattern cut (at θ = θοί; Ψ = Ψj) after radome installation.

It should be stressed once again that it refers to calculated errors in the coordinate system of PAA phasing angles (αх; αy). According to the ratios known from the mathematics (metrology), error of a value representing a complex function (of multiple variables) is determined by calculation of its differential through the differentials of its arguments. In the considered case the arguments would be: ∆θoi, determined by mean-square error of PAA beam setting and error of ARSD setting in the azimuthal plane; Δθοi·, Δθ yi, which will be determined both by random phasing errors and errors in measuring the coordinates of the minima of difference RP (within the algorithm accepted); ∆Ψj, determined by random error of ARSD setting by heel angle.

It follows from all of the above that for a general case of measurements in the oblique planes even the error of arguments being measured is indeterminate, while the task of assessing the computing error of functions Δαxi j , Δαyi j themselves by the mathematical methods becomes a challenging one and goes beyond the authors’ competence.

However, having collected measurement statistics, the authors would be quite able to asses that confidence interval which accommodates all measured implementations, given the influence of all random factors in the measurements made. Unfortunately, within the framework of this paper the said assessment was not performed for the lack of time, but it can be made in the course of further investigations in this direction. In practice, for the proposed compensation method it might look as follows.

Let us consider the measurements taken on one of the sections at heel angle Ψ = Ψj. The proposed method of compensation prior to installing RPR implies creation of a corrected data array with phasing coordinates at all possible angles θοί,, where the conditions for boresight direction θai =θyi are satisfied. Later on, after RPR installation and performance of respective measurements, those coordinates will be used in calculation of the ABE components. In this way, if we make several independent measurements

before RPR installation and then the same number of measurements after installation, then the calculation program (within the developed Stend BEK SW package) can be made to compute ABE components in a criss-cross pattern, combining arbitrary data before RPR installation with those obtained after it. For example, it can be visualised as a criss-cross merging of the measured arrays (Fig. 12). Thus, with three measurements (see Fig. 12) taken before and after, we have 9 implementations of each one of the ABE components.

In Fig. 13, these 9 calculated implementations (e. g., components Δαxi, j), presented together, form some kind of a statistical “corridor”, which may be laid in the base of a confidence interval of measurements.

To perfect the method and compare the obtained results with the measurement data from other methods, a model of RPR simulator was proposed and manufactured, having the following properties:

Shown in Fig. 14 is a manufactured simulator mockup. It had just one (vertical) symmetry plane and successfully fulfilled its main function: to introduce ABE when installed on the tested PAA.

The plots of ABE components obtained with the use of the proposed method for the manufactured RPR simulator mockup at some angles of PAA – RPR system setting by heel are given in Fig. 15. The frequency on which measurements were taken corresponded to the wavelength of 3 cm. All measurements made with the RPR simulator mockup are represented on a single frequency band letter. The plots of ABE components (∆αхi, j and ∆αуi, j) in angular minutes in coordinates θ, Ψ at system heel angles Ψ = ±30°, ±45°, ±60° are given in Fig. 15–17.

The symmetry plane for this RPR simulator mockup, schematically shown in Fig. 18, will be the vertical plane passing through its longitudinal axis. The RPR surface points, arranged symmetrically relative to it (some of them are shown in Fig. 18) apparently must have similar properties in terms of capability to refract the incident electromagnetic wave. As follows from simple representations of geometrical optics, for component Δαх (∆Аz – azimuthal component in Fig. 18), it is a match by absolute magnitude and opposition by sign. For component Δαy (∆El – elevation component in Fig. 18), it is a match by absolute magnitude and sign. The plots given below demonstrate this vividly enough. It is for this reason that the measurement results are represented by paired plots at respective angles ±Ψ.

The absence of full even or odd symmetry is partly explained by considerable flaws in manual manufacture of the mockup.

Measurements of individual ABE components (Δα xi, j and Δα yi j), were performed on the RPR simulator for some of its section (i. e., at different heel angles of the antenna – radome system), with their results combined in the plot and presented in Fig. 19. Smooth evolution of component ∆αх while proceeding from angles Ψ = +60° to the limit heel angle Ψ = 0° can be traced in Fig. 19 (а) and for component Δαу – from Ψ = 30° to angle Ψ = 90°. in Fig. 19 (b). Therewith, at system setting angles Ψ = 0° and Ψ = 90°, where the proposed method does not yield a result any more because of degeneracy of one of the difference RP’s minimum, the components were measured by the classical difference method. The latter implies deduction of the measured angular coordinates of respective difference RP before and after RPR installation.

Besides, for ABE component ∆αх in Fig. 20 and ABE component ∆αу in Fig. 21 at system heel angle Ψ = ±60°, the data obtained according to another method are given. The method consists in immediate measurement of the coordinates of spatial minimum formed by difference dynamic spatial RP (DSRP) of the tested PAA [

The described method for measuring ABE in the PAA angular region accessible for scanning allows to determine both error components conditioned by the presence of a radome. It is distinguished by simplicity of implementation and mathematical processing.

In the blind sectors close to the orthogonal planes (0° and 90°), where signal minimum of one of the difference channels drifts and becomes unavailable for determining its angular coordinate, this method stops functioning. For this reason, to have full advantage of it, it is crucial to avail of a good tool, i. e. a PAA with the depth of zeros of the shaped difference RP no worse than 30–35 dB. The sector of measurements by Ψ, where the difference RP minimum (given theaccepted algorithm of its measurement) could be unambiguously determined for the PAA available to the authors, was limited by angles –30°…–70° and +30°…+70°. The ABE components in blind zones inaccessible for measurements by the method in question can be determined, for example, using the method of extrapolation as per the adjacent measured zones or augmented by the data obtained in those sectors using other methods. A promising method could turn out to be that of electronic shaping of special oblique difference RP. It should be pointed out that in two limit cases of PAA – RPR system installation, Ψ = 0° or Ψ = 90°, formulas (17), (18) for calculation of one of the ABE components degenerate into elementary expressions by which the ABE are calculated using the classical difference method. Hence, in the two limit cases, one of the ABE components is calculated by a well-known method. The proposed method requires further thorough metrological study for its possible subsequent attestation and use as a method for determining the characteristics of a PAA (APAA) – RPR system (ABE, radome TF, etc.).

The authors declare that there are no conflicts of interest present.