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Method of forming and scaling phased array expanded beams
https://doi.org/10.38013/2542-0542-2019-3-19-29
Abstract
For citations:
Gribanov A.N., Gavrilova S.E., Pavlovich O.V., Moseychuk G.F., Titov A.N. Method of forming and scaling phased array expanded beams. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2019;(3):19-29. https://doi.org/10.38013/2542-0542-2019-3-19-29
Introduction
Contemporary airborne radars often employ antennas represented by passive phased antenna arrays (PAA), which must comply with the requirement to promptly change beam shape and parameters. Amplitude distribution of the PAA is determined by the distribution system parameters and normally cannot changed in regular service conditions. All beam shape transformations can only be performed through phase distribution (PD) control. Thus, for prompt formation of a beam with specified parameters it is necessary to apply such phase synthesis methods that make it possible to find solutions within a short period of time; therefore, the applied synthesis algorithm shall imply as few simple mathematical operations as possible.
The algorithms of prompt PD finding considered in this paper enable to solve the tasks set to a large extent. They are based on the method of phase synthesis of fan partial radiation patterns (FPRP) [1][2][3]. Lying in the core of the FPRP method is a simple ratio for a two-element antenna array between the direction of the radiation pattern (RP) maximum and the phase difference on the radiators [4]. The conditions of partial RP superposition and equality of the radiated energy, calculated across the array and in the angular space of the formed beam, are used as well. Due to clarity and simplicity of the approaches applied and the algorithms obtained, the phase synthesis operation comes down to elementary algebraic operations which can be performed immediately in the process of radar operation. Modified versions of the method also enable to implement phase synthesis of volumetric RP of flat PAA with arbitrary aperture shape [5] and to serve as the base for finding optimal solutions for RP phase synthesis in accordance with the specified optimisation criterion [6].
It should be noted that the FPRP method has features in common with the geometrical optics (GO) method. The mathematical apparatus of the GO method is based on asymptotic estimate of the Fourier integral with rapidly oscillating phase function, which is called stationary phase method in mathematics [7]. The general mathematical apparatus of the GO method [8][9][10] is quite complex, it uses the condition of equality of the excitation and radiation energies, widely applies the notion of a stationary point and allows to take into account fields radiated by the antenna edges. The GO method is a universal synthesis tool, enabling in some particular cases of linear PAA to obtain relatively simple phase solutions.
As opposed to GO, the FPRP method is based on very simple provisions. It can be regarded as an engineering method of phase synthesis which, as concerns discrete apertures, provides solutions similar to the results obtained by the GO method. At the same time, it is exactly simplicity and clarity of the FPRP method that have enabled to discover the property of scaling synthesis solutions, using which it is possible to obtain a multitude of other options by multiplying phase distribution by the scaling factor [11]. This property allows additional reduction of the time of operational change of the expanded PAA beams.
This paper discusses the potential of the FPRP phase method when applied for one-dimensional expansion of a flat PAA beam, as well as specific features of using phase synthesis solutions scaling. The basic concepts and mathematical expressions used in the FPRP method are examined in detail, since the main features of scaling operations stem directly from them.
Basic concepts and mathematical relationships of the FPRP method
The concept of the method of fan partial radiation patterns is based on representation of a PAA aperture, having М radiators, in the form of a subarray population (М - 1) [1][2][3]. Each of the newly formed subarrays consists of two radiators. Any internal radiator of the aperture is conventionally split in two, each of the parts having the same coordinates and half-values of the amplitudes relative to the initial level. Such virtual halves of the radiators, separated from one another to a distance equal to the array spacing, form two-element paired subarrays. Fig. 1 shows formation of paired subarrays for a radiator line.
Fig. 1. Splitting of linear PAA into paired subarrays
Each one of such subarrays forms a partial RP, whose angular position can be controlled by superimposing phase shift on its radiators. To obtain an expanded beam of the entire PAA, it is necessary to position the peaks of those beams at a certain angular distance from one another (Fig. 2).
Fig. 2. Expanded beam formation
Lying in the core of the method are simple energy relations. During formation of expanded beams, all the energy radiated by the antenna array shall be distributed within an angular space from umin = sinθmin) до Mmax = sinθmax), and the equation of energy balance for a PAA with linear discrete aperture can be translated as
where b - normalisation factor;
М - number of radiators in aperture;
(М - 1) - number of paired subarrays, with the same number of angular intervals of beam splitting;
A2m_ср - radiated power mean value on the aperture segment а Δxm;
Δxm - length of the m-th antenna aperture segment;
F2m_ср - mean value of the radiated power density in angular interval Δum;
Δum - m-th angular interval of beam splitting, expressed in variables u = sin(θ).
If the levels of amplitude distribution and the formed beam are coherent in energy, then b = 1. Otherwise, the value of b is easily determined from the results of integral energies normalisation.
From the selected aperture geometry and the selected current excitation on the elements, all terms of the left-hand sum become known. The desired beam shape F(u) is considered set, with angular coordinates {Δum} and their respective mean values {F2m_cp} remaining unspecified.
Equation (1) has a set of solutions. The simplest one can be obtained under term-by-term equality, i. e., when the following relationship is implemented for all m values
Physically, expression (2) means that the energy radiated by each m-th aperture segment located on interval ∆xm, must be directed into the respective m-th angular interval, and it is
this energy that determines the energy density in interval Δum. Fig. 3 clearly demonstrates determination of the position and dimensions of the m-th angular zone when using respective integral functions P(x) and P(u), which express the energy balance of energy distribution in the PAA aperture on the one side and in the angular space on the other. The desired beam shape is ensured due to correct positioning of partial beams with account of their level.
Fig. 3. Expanded beam formation
As shown above, it is convenient to use two adjacent radiators in the PAA as subarrays. The level of peaks in partial RP {Fm (0)} is determined by the type of amplitude distribution in the array and the location of the corresponding pair of radiators:
The width of angular interval allocated to each partial beam (Figs. 2, 3) is proportional to the partial beam level and is determined in the most elementary case of sector-wise expansion of the initial beam of an equidistant PAA by the expression
Partial beam peaks shall be directed into the centres of the respective angular intervals, therefore the peak directions are calculated by the formula
For partial beams positioning in direction {um} it is necessary to determine phase shift on the paired subarray radiators to a value
∆φm+1 = -kdum (6)
Obviously, the phases of split radiators (common radiators of the adjacent subarrays) are to be equal (see Fig. 1). Considering that the phase of the first radiator (m = 1) can be left unchanged (or taken equal to zero), the sought phase of other radiators with numbers m > 1 will be determined by the formula
It is the obtained phase distribution {φ m} that will be forming a beam of desired shape F (u). Applying the method considered, it is possible to form both simple one-dimensionally expanded sector beams and more complex ones, for example, of type cosec2 (θ).
It is notable that, with the direction of all partial beams {um} changed by one value Δu0 the formed beam shifts by this angular value, which corresponds to addition of linear phase distribution along the aperture
The quality indices of expanded beams (oscillations amplitude in the beam region, width of the beam slopes, side lobes level, etc.) depend on the amplitude distribution parameters (primarily on the field level at the edge), aperture shape, number of radiators, their spacing values, desired beam shape and parameters, etc. These dependencies have a complex nature and their identification, even for a PAA with selected geometry and excitation, requires fairly extensive research.
The main advantage of the method consists in the use of simple algebraic expressions for calculation of phase distribution, which yield an unambiguous solution.
Scaling of expanded beams
The scaling property consists in the presence of certain ratios between phase distribution and beam width. By the scaling property we shall imply the following: if primary PD φ(χ) in the aperture of a linear PAA forms a beam expanded α -fold, and secondary PD βφ(χ) forms a beam expanded αβ -fold, then the PD and beam width have scaling property by variable u = sin (θ) with factor β.
Given below are definitions and arguments concerning several important properties of the FPRP method which are very useful in synthesising expanded beams and carrying out their scaling.
Lemma 1. If the PD that forms an expanded beam is formed according to the FPRP method, then the PD and beam width are linked by the scaling property.
If primary PD, which expands the beam according to the FPRP method α -fold, is already formed at the PAA aperture, then directions {uam} of the partial beams and phase shifts on each binary subarray {Δφαm} are determined. Additional change of beam width β-fold means that the partial beams must be positioned in directions {βuαm} (the fan of beams (Fig. 2) is folding or unfolding depending on the value of β). Hence, in accordance with (7), new PD on the m-th radiator equals to
It means that for additional change of the beam width β -fold, it is necessary to additionally change the PD β -fold. In this way, it is proved that the PD and beam width, as formed according to the FPRP method, possess the scaling property.
Lemma 2. If the PD and beam width are linked by the scaling property, then for the scaling operation in a general case it is necessary to use the primary PD not reduced to interval 2π.
For argumentation, let us take an arbitrary aperture radiator with number m. According to the FPRP method, the radiator has a phase which can be expressed as two terms, one of which is integer 2π and the other - remainder less than 2π:
Ψm = 2πn + φm . (10)
When carrying out scaling operation, i. e., additional expansion of the beam into arbitrary number β, the PD must be changed β -fold:
βψ m =β2πn + βφ m. (11)
Hence, when scaling a reduced PD, i. e., if the term with integer 2π is cast out, in a general case we obtain a wrong PD, and scaling will be inefficient. It is notable that in a particular case of scaling an integer number of times (β = 2, 3, 4, etc.), the beam will be additionally expanded predetermined-number-fold even if a reduced PD is used. It is also obvious that the beam will be additionally expanded correctly if the changing range of PD obtained at the primary scaling does not exceed 2π (option n = 0).
Lemma 3. If the PD and beam width are linked by the scaling property, then during the operation of scaling β -fold, the boundaries of beam start and end are changing β -fold by variable u = sin(θ).
It follows from expressions (9) that in the operation of scaling β -fold, directions u of all partial beams (including start and end ones) are changing β -fold. In accordance with the theory of forming expanded beams by the FPRP method, the boundaries of the start and end partial beams determine the boundaries of the formed beam start and end. This property has a general character and does not depend on the expanded beam shape.
Lemma 4. If the PD and beam width are linked by the scaling property, then during the operation of scaling β -fold of a beam primarily expanded α -fold, the same PD and beam shape are formed that were obtained at the primary beam expansion αβ -fold.
This property follows immediately from expressions (9), which are also used in primary beam formation according to the FPRP method (7).
When using the FPRP method and the scaling property, it is also necessary to take into account the following considerations:
- beam scaling factor means beam width change by variable u = sin(θ);
- as beam expansion factor, it is possible to use the parameters of expansion by different levels (–3 dB, –6 dB, etc.). It is necessary to consider that at the top of an expanded beam there are oscillations, and at the low levels there are side lobes “stuck” to the beam;
- the FPRP method and the scaling property make it possible to ensure a desired shape of the amplitude RP in the beam expansion region, but they do not determine the phase RP;
- the expanded beam shape will not change if the sign of synthesised or scaled PD is inverted or a constant component is added to it;
- with addition of a PD which is changing linearly along the aperture, the synthesised beam shifts in the angular region without changing its shape. If a shifted beam is subjected to scaling, its boundaries will change in accordance with Lemma 3;
- the FPRP method and the scaling property are described in this paper as applies to the directivity multiplier (DM) of a multi-element PAA, i. e., for the case of using omni-directional independent radiators. For real PAA, the described theory holds true when synthesising and scaling beams in the angular region, where the RP of one radiator (scanning pattern) is close to unity. When synthesising broader beams, the desired beam shape has to be corrected with account of the scanning pattern, appropriately increasing levels of the synthesised beam in remote angular regions.
It is necessary to point out that, apparently, solutions obtained by the GO method possess the scaling property as well. However, this presumption and the degree of commonality of its possible use need to be proved. At least the solutions obtained for linear PAA in [9] do have such property.
Results of mathematical simulation of sector beams
The scaling properties are easily verified by the mathematical simulation means. Fig. 4 shows 5 options of phase expansion of the beam. The simulation results were obtained for a linear PAA comprising 40 radiators, arranged with spacing 0,5λ, and having amplitude distribution of cosine-squared-on-a-pedestal type, with field level at the edge equal to 0.25.
Shown in Fig. 4 in red is phase distribution applied for the primary beam expansion, and the corresponding expanded beam. Using scaling factors β = {1,7, 1,72 - 2,9, 1,73 - 4,9, 1,74 - 8,4}, we obtain respective expansion options of the initial sector beam. The greater the range of PD change in the aperture, the wider the formed beam.
The proved scaling properties easily extend to PAA with flat apertures. In this case it is necessary to use an equivalent linear aperture as a linear PAA [4].
Given in Fig. 5 are examples of sector beams formation in a flat PAA in the elevation plane with the use of the methods being developed. Shown in green colour are the solutions (PD and RP) of beam primary expansion 8-fold. The next 2 secondary solutions were obtained by 1.5- and 1.52-fold scaling. For comparison, Fig. 5 (d) also shows the initial RP, and in Fig. 5 (c) an example of a 3D RP with sector beam is given.
With the scaling property available, it becomes possible to considerably simplify the process of forming a family of expanded beams in flat PAA in the inclined planes as well. This possibility can be implemented through changing phase distribution formation plane in the aperture. The latter is especially relevant, for example for aircraft-mounted PAA in situations when it is necessary to stabilise beam position and shape in the angular space during aircraft evolutions.
Results of mathematical simulation of beams of cosec2(θ) type
An indisputable advantage of the FPRP method is the possibility to scale beams of a more complex shape, like those of the cosec2(θ) type (Fig. 6).
Given in Fig. 6 (a, b) are the PD forming cosecant beams. Blue colour designates the primary PD, and red and green - those forming an expanded and a narrowed beam, respectively. Given in Fig. 6 (d) are the results of cosecant beams scaling 1.7-fold. Having additionally built the graphs of functions cosec2 (θ) = 1/sin2(θ) it is easy to discover correspondence of cosecant beams formed by the FPRP method to respective mathematical functions, which actually confirms a possibility for scaling beams of the cosec2(θ).
It is important to consider that, when scaling a cosecant beam β -fold in accordance with Lemma 3, the beam boundaries change according to law unew = βuold. It means that the start and the end of a scaled beam are to shift to different angular distances, which can be seen in the figures given.
It must be pointed out too that according to Lemma 4 it is irrelevant by which technique (scaling or immediate synthesis by the FPRP method) the presented expansion options were obtained. The parameters of the expanded beams will be identical. Likewise irrelevant is which of the expansion options is used for scaling as the primary one.
Results of experimental research
Given in Fig. 7 are the results of experimental research in forming sector and cosecant beams whose phase solutions were obtained with the FPRP method applied. The measurements were performed using a PAA with round aperture, having amplitude distribution tapering along the radius, with aperture efficiency 0.91 and PD implementation random error with RMS deviation σφ ≈5°. Notably, the level of the peak side lobes for this PAA under cophased distribution being formed is ≈ –28 dB.
It follows from the data presented that the accuracy of the simulation and measurement results for RP levels over –28 dB reaches a fraction of decibel. Across the operating range, the maximum difference of the simulated and measured cosecant RP from the cosec2 (θ) function is ≈1.5 dB.
Conclusion
The paper considers the FPRP method due to which the operation of phase synthesis of one- dimensionally expanded beams of specified shape can be performed for a PAA. For the first time a new scaling property is studied, enabling to perform the operation of additional expansion of PAA sector beams based on a previously found expansion option. In this way, if it is necessary to implement in a PAA a family of expanded beams of the same type, it suffices to find, using the FPRP method, one primary PD, which expands the beam α -fold, and obtain all other beams with αβ -fold expansion by multiplying the primary phase distribution by factor β.
Four important properties are formulated and proved, which must be taken into account when scaling.
The results of mathematical simulation for forming one-dimensionally expanded sector and cosecant beams have proved performance capability of the FPRP method and scaling efficiency of the obtained solutions.
Availability of the scaling property provides considerable advantages in using phase synthesis of expanded beams by the FPRP method, the most important of which are as follows:
- possibility to unambiguously determine primary phase distribution based on the selected amplitude distribution in the aperture and the desired beam shape, applying simple algebraic operations;
- simplicity of the mathematical operations of scaling, consisting in multiplication of the primary phase distribution by the scaling factor, determines the ease of its practical implementation in PAA;
- possibility to form beams with desired expansion factors immediately during operation of radar with PAA;
- possibility of application in active and passive PAA with linear and flat apertures.
There are reasons to believe that phase solutions obtained by the GO method possess the scaling property as well.
The discussed method of phase synthesis with the use of the scaling property has passed experimental verification in the Joint Stock Company V. V Tikhomirov Scientific Research Institute of Instrument Design on the samples of developed PAA and APAA during formation of sector and cosecant beams.
About the Authors
A. N. GribanovRussian Federation
S. E. Gavrilova
Russian Federation
O. V. Pavlovich
Russian Federation
G. F. Moseychuk
Russian Federation
A. N. Titov
Russian Federation
Review
For citations:
Gribanov A.N., Gavrilova S.E., Pavlovich O.V., Moseychuk G.F., Titov A.N. Method of forming and scaling phased array expanded beams. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2019;(3):19-29. https://doi.org/10.38013/2542-0542-2019-3-19-29