# A method for measuring angular bearing errors in the “antenna - radome” system in the phased-array beam scanning region

### Abstract

### Keywords

#### For citations:

Makushkin I.E.,
Dorofeev A.E.,
Gribanov A.N.,
Gavrilova S.E.,
Sinani A.I.
A method for measuring angular bearing errors in the “antenna - radome” system in the phased-array beam scanning region. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2019;(2):7-24.
https://doi.org/10.38013/2542-0542-2019-2-7-24

## Introduction

Errors in determining antenna boresight position (angular bearing errors, ABE) occurring in the presence of a radioparent radome (RPR) [1] shall be taken into account during operation of an airborne radar (ABR) [2]. Works focusing on ABE measurement in two orthogonal planes only, in azimuth or elevation, are quite frequent [3, 4]. At that, the ABE is determined as a difference between the measured coordinates of zeroes of respective difference radiation pattern (RP) before and after installation of the radome (difference method). Such an approach gives an idea of one of the ABE components only. A randomly deflected beam of phased array antenna (PAA) in the antenna – radome system acquires angular bearing errors by both coordinates at once (Fig. 1) [1]. Therefore, the development of methods for practical assessment of radome influence on the drift of “zeroes” of the boresight characteristics in the entire beam scanning region is a vital task of the present day.

**Fig. 1**. Randomly deflected beams of phased antenna array (PAA) in system “antenna – radome”: 1 – normal to aperture; 2 – possible setting positions of PAA beam; 3 – PAA aperture; 4 – PAA scanning cone; 5 – RPR surface

## Geometry of the task for one section of RPR surface

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 Ψ_{j–} (pos. 5). Under arbitrarily selected spatial angle θ_{i– }(pos. 7) of deflection from the normal to the antenna aperture (pos. 8), controlling angles L_{x} and L_{y} (pos. 3 and 4) between respective axes X, Y and beam setting direction (pos. 6) are calculated by the formulas

cos L_{x} = sin θ_{i} cos Ψ_{j}; (1)

cos L_{y} = sin θ_{i }sin Ψ_{j}, (2)

and the array phasing angles (α_{х}; α_{y}) transmitted from ABR computer to the PAA beam steering unit, by the formulas

α_{х} = 90 - L_{x}; (3)

α_{у} = 90 - L_{y}. (4)

**Fig. 2**. Geometrical representation of the task for determining ABE components: 1 – PAA; 2 – PAA aperture centre; 3, 4 – steering angles L

_{x}and L

_{y}; 5 – arbitrary PAA heel angle Ψ

_{j}; 6 – arbitrary direction of PAA beam setting; 7 – angle θ

_{i}between normal to aperture and direction of PAA beam setting; 8 – normal to PAA aperture (beam set along normal has θ

_{i}= 0); 9, 10 – RPR axis and forming surfaces; 11 – azimuthal rotation axis of PAA installed on ARSD; 12 – PAA beam setting plane; 13 – discrete points of PAA beam setting; 14 – ARSD azimuthal turn plane; 15 – section line on RPR surface by which ABE components are measured

Given such calculation of phasing angles α_{х}, α_{у} and with system heel angle Ψ_{j} being constant, any angle θ_{i} 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 (−θ_{i}), 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 (−θ_{i}), the coordinates of the cut minima of both difference patterns can be changed simultaneously.

## Mathematical apparatus for calculating angular bearing errors

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.

**Fig. 3**. Spatial difference azimuthal (a) and elevation RP (b) on the plane (u, v)

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 L_{x}, V = cos L_{y}. L_{x} и L_{y} – 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 θ_{o} and the angle of PAA setting by heel Ψ. Two characteristic points can be singled out on plane (U, V).

**Fig. 4**. Cut lines of spatial difference RP (Az – green cut lines, El – red cut lines) in the coordinate system of guiding (steering) cosines cosL

_{x}, V = cosL

_{y}for a PAA set to heel angle Ψ

In point 1, before RPR installation, a PAA beam is set, according to formulas (1)–(4), as per spherical coordinates θ_{0}, 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 (u_{0}, V_{0}). 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 θ_{α} = θ_{у}, coinciding with beam deflection angle θ_{0}.

**. Screenshots of measuring complex display under precise setting of bearing to signal source (a) and in the absence of it (b)**

Fig. 5

Fig. 5

In point 2, after RPR installation, the PAA beam is still set as per spherical coordinates θ_{0}, 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 (U_{1}, V_{1}) 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 θ_{0}, Ψ the actually measured minima of the cuts of both spatial difference RP, determined by the end of vector sin θ_{α} and sinθ_{у}, will be fixed at angles θ_{α }and θ_{у}, respectively. In that case, the fixed angles of the measured minima θ_{α }and θ_{у} are not equal and do not coincide with θ_{o} (Fig. 5, b).

On the plane of the guiding cosines, the coordinates of point 1 are U_{0} = cos L_{x}, V_{0} = cos L_{y}, 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 (-θ_{ο}) and measuring angles θ_{α} and θ_{у }(the initial setting angle θ_{o} and Ψ are known), which changed after RPR installation, and also taking into account that

u_{1} = u_{0} + du; (5)

du = (sin θ_{0} - sin θ_{α} )cos Ψ; (6)

V_{1} = v_{0} + dv; (7)

dv = (sin θ_{у} - sin θ_{0} )sin Ψ, (8)

we calculate du, dv, by formulas (6), (8) and u_{1}, v_{1}. by formulas (5), (7).

Considering that

L_{хo} = arccos u_{0}; (9)

L_{x1} = arccos u_{1}; (10)

L_{yo} = arccos v_{0}; (11)

L_{y}_{1} = arccos V_{1}, (12)

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

Δα_{х} = L_{x1} - L_{хo}. (13)

Whereas the increment of component Δα_{y} to the initial phasing angle of the PAA being tested by coordinate α_{y} , caused by RPR installation on the PAA, we define as

Δα_{y} = L_{y1} - L_{y0}. (14)

In fact, it is Δα_{х} and Δα_{y} 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 (θ_{0}, Ψ) which corresponds to the array phasing coordinates α_{х} and α_{y} . 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 Δα_{х}, Δα_{y}, as a function of arbitrary (permissible for the PAA being tested) spherical coordinates of system installation θ_{0ί}, Ψ_{j}.

In this case, boresight error components Δα_{xi, j}, j, Δα_{yi, j} proper, introduced by the RPR at angles θ_{0} = θ_{0ί}, Ψ = Ψ_{j} of the antenna – radome system installation in the variables of spherical coordinate system of the antenna can be calculated by the formulas

where Δα_{xi, j} – component of the boresight error introduced by radome by angle α_{х} (at θ = θ_{οί}, Ψ = Ψ);

θ_{αί} – actual measured angle of the minimum of difference spatial azimuthal pattern cut (at θ = θ_{0ί}, Ψ = Ψj) after radome installation;

Ψ_{j} – current value of antenna system heel angle relative to the ARSD azimuthal scanning plane;

θ_{oi} – current value of angle θ_{o} 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;

Δα_{yi,j} – component of the boresight error introduced by radome by angle α_{у} (at θ = θ_{oi}; Ψ = Ψ_{j});

θ_{уi} – actual measured angle of the minimum of difference spatial elevation pattern cut (at θ = θ_{oί}; Ψ = Ψ_{j}) after radome installation.

It should be stressed once again that it refers to calculated errors in the coordinate system of PAA phasing angles (α_{х}, α_{y}). 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 θ_{αί} = θ_{yi} 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 θ_{αί} and θ_{уi}. are measured. In this case the proposed mathematical model is functioning, and the boresight error components Δα_{xi},_{j} и Δα_{yi},_{j} can . be determined by formulas (15)–(16) from the magnitudes and signs of θ_{ai} and θ_{уi} deviations from the initial θ_{oi}, where condition θ_{αί} = θ_{yi}, 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.

## Geometry of the task for a set of sections over entire RPR surface

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 Ψ_{j} and beam deflection angles θ_{oi}. 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 (Δα_{xi},j and Δα_{yi},j) are measured are shown in different colours (see Fig. 6).

**Fig. 6**. Section lines of arbitrarily shaped RPR surface and points on it by which ABE components are measured

## Principle of electronic compensation of PAA beam initial setting in ABE measurement

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 θ_{0} = θ_{oi}, Ψ = Ψ_{j}.

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.

**Fig. 7**. Schematic representation of ARSD used in measurements: 1 – horizontal (adjustable) ARSD platform; 2 – adjustable lifting support of ARSD horizontal platform; 3 – ARSD azimuthal rotary platform with structural elements of vertical supports; 4 – azimuthal turn axis; 5 – axis of tested PAA turning by heel; 6 – axis of tested PAA turning in the elevation plane; 7 – tested PAA aperture plane; 8 – normal to tested PAA aperture plane; 9 – mismatches by heel angle ∆ψ; 10 – mismatches by beam deflection angle ∆θ; 11 – conventional vertical planes, equidistant from tested PAA phase centre; 12 – conventional plane of azimuthal turn of ARSD with tested PAA for the case ∆ψ = 0, ∆θ ≠ 0 (blue line); 13 – conventional true horizon line (green line); 14 – conventional plane of azimuthal turn of ARSD with tested PAA for the case ∆ψ ≠ 0, ∆θ = 0 (red line)

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 α_{xi,j} and α_{yi,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 θ_{α} = θ_{y }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 θ_{o} = θ_{oί}, Ψ = Ψ_{j} 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 θ_{o} = θ_{οί}, Ψ = Ψ_{j} 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 θ_{a} and θ_{y} and calculate additives ∆α_{xi, j комп} и ∆α_{yi, j комп} by formulas (15), (16), the only difference being that, taken with the respective sign and added to the initial phasing coordinates α_{xi}, _{j }_{нач} и α_{yi}, _{j }_{нач} (at beam setting angles θ_{ο} = θ_{oi}, Ψ = Ψ_{j}), they will act as the compensating ones and are supposed to bring the system to a point where condition θ_{α} = θ_{y }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 Δα_{xi} _{j} and Δα_{yi} _{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 θ_{a} and θ_{y} (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 θ_{o} = 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.

**Fig. 8**. Data of practical measurements of θ

_{а}and θ

_{y}(before RPR installation) as per signals of two difference channels under initial setting of tested PAA by heel Ψ = 60° and angle of beam deflection from normal θ

_{о}= 45° before (а) and after (b) compensation

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 ∆α_{xi, j комп} and ∆α_{yi, 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.

**Fig. 9**. Block diagram of antenna-collimator complex (AKK1-12)

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.

**Fig. 10**. Stend BEK program interface

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 Ψ_{j} and by the generated job file for θ_{oi} , and, from the coordinates of the initial divergence θ_{a} and θ_{у},

compensating additives in each point of PAA initial phasing (∆α_{хi, j комп} and ∆α_{уi, 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 α_{х},_{j} and α_{у},_{j} , ABE components (Δα_{χι}·,j и Δα_{yiJ}), 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.

## Assessment of measurement errors

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 θ

_{oi}, Ψ

_{j}) 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 θ

_{oi}, Ψ

_{j}, 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).

**Fig. 11**. Method for compensating errors introduced by ARSD mechanical system in elevation channel drive at ∆El

_{теод}= 3 ang. min, ∆El

_{ас}= 4.12 ang. min

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 – θ_{oi} = 0 (more exactly, the azimuthal angle of the found electrical axis at heel angle Ψ_{j}). 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 – Ψ_{j}. 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 θ_{oi}. Control of object (PAA – RPR) positions throughout the entire range of setting angles θ_{oi}, Ψ_{j} 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 (Δα

_{xi},

_{j}Δα

_{yi},

_{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 Δα_{xi,j} – component of the boresight error introduced by radome by angle α_{х} (at θ = θ_{oi}; Ψ = Ψ_{j});

Ψ_{j} – current value of antenna system heel angle relative to the ARSD azimuthal scanning plane;

θ_{oi} – 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;

∆α_{уi, j} – component of the boresight error introduced by radome by angle α_{у} (at θ = θ_{οί}; Ψ = Ψ_{j});

θ_{ai} – actual measured angle of the minimum of difference spatial azimuthal pattern cut (at θ = θ_{οί}; Ψ = Ψ_{j}) after radome installation;

θ_{yi} – actual measured angle of the minimum of difference spatial elevation pattern cut (at θ = θ_{οί}; Ψ = Ψ_{j}) after radome installation.

It should be stressed once again that it refers to calculated errors in the coordinate system of PAA phasing angles (α_{х}; α_{y}). 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: ∆θ_{oi}, determined by mean-square error of PAA beam setting and error of ARSD setting in the azimuthal plane; Δθ_{οi}·, 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); ∆Ψ_{j}, 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 Δα_{xi} _{j} , Δα_{yi} _{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 = Ψ_{j}. The proposed method of compensation prior to installing RPR implies creation of a corrected data array with phasing coordinates at all possible angles θ_{οί},, where the conditions for boresight direction θ_{ai} =θ_{yi} 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.

**Fig. 12**. Statistical processing of data arrays on the basis of measured data merging

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

**Fig. 13**. Obtaining a confidence interval of calculated implementations (components ∆αх

_{i, j})

## Use of a quick-release RPR simulator mockup with “artificially” introduced inhomogenuities

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:

- real capability, observed during measurements, to refract incident plane wave front (i. e., introduce ABE by both components ∆α
_{хi, j}and ∆α_{уi, j}); - small weight, to ensure minimum mechanical impact on the ARSD drives with the PAA – RPR system installed on it;
- simple installation and removal of the simulator mockup with the tested PAA installed on ARSD, with the minimum possible impact on its mechanical drives;
- different areas of the simulator surface shall have different refraction coefficients (as in a real RPR).

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.

**. RPR simulator model**

Fig. 14

Fig. 14

## Results of ABE measurements performed on RPR simulator mockup

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 (∆α_{хi, j} and ∆α_{уi, 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 Δα_{х} (∆Аz – azimuthal component in Fig. 18), it is a match by absolute magnitude and opposition by sign. For component Δα_{y} (∆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.

**. Schematic representation of arbitrary RPR and characteristic points on its surface**

Fig. 18

Fig. 18

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 Δα_{у} – 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 ∆α_{у} 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 [5], before and after RPR installation, when the position of this minimum changes. Measurement of difference DSRP was performed at different setting angles of the tested PAA to the plane front of the incident electromagnetic wave from the collimator. The attribute “dynamic” in the DSRP abbreviation is applied in the sense that spatial RP themselves were measured in a certain region of adjustments by θ, Ψ relative to the initial setting due to purely electronic PAA beam scanning in that region. For the sake of convenience, ABE components measured on the RPR simulator model with the use of DSRP and obtained by the compensation method at section Ψ=±60°, are given in the plots together. Due to complexity of practical implementation in the measuring complex and immense volume of the data obtained in the process, this method is at the development stage as yet. However, good repeatability of data obtained on the same RPR simulator mockup using two different approaches holds out a hope that the method of ABE measurement with the use of DSRP will find its place as well.

## Conclusion

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 the

accepted 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.).

### About the Authors

**I. E. Makushkin**Russian Federation

**A. E. Dorofeev**Russian Federation

**A. N. Gribanov**Russian Federation

**S. E. Gavrilova**Russian Federation

**A. I. Sinani**Russian Federation

### Review

#### For citations:

Makushkin I.E.,
Dorofeev A.E.,
Gribanov A.N.,
Gavrilova S.E.,
Sinani A.I.
A method for measuring angular bearing errors in the “antenna - radome” system in the phased-array beam scanning region. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2019;(2):7-24.
https://doi.org/10.38013/2542-0542-2019-2-7-24