# Generalized characteristics of expanded sector beams formed by phased antenna arrays obtained by scaling phase distributions

### Abstract

#### For citation:

Pavlovich O.V., Gribanov A.N., Gavrilova S.E., Moseychuk G.F., Kuznetsov I.A. Generalized characteristics of expanded sector beams formed by phased antenna arrays obtained by scaling phase distributions. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2020;(1):46-54.
https://doi.org/10.38013/2542-0542-2020-1-46-54

## Introduction

The main specifics of phased antenna array (PAA) operation as part of a multi-function radar consist in enabling prompt change of emission characteristics. This can only be implemented through the control of emitters’ excitation phases at a known and fixed amplitude distribution (AD). There is a significant number of theoretical papers on phase synthesis of beams formed by PAA, e. g., [1-5]. Most of them imply availability of element excitation phase sets pre-calculated and stored in the PC memory for all kinds of radar operation modes. At the same time, the high-speed performance of the state-of-the-art computing facilities being part of PAA allows to calculate phase distribution (FD) in real time directly based on the high-speed algorithms. At that, the operations used for new phase values calculation should be simplified to the maximum. This paper considers the method of fan partial radiation patterns (FPRP), which allows many expansion options to be obtained on the basis of one expansion option by multiplying the phase shift values of the signals from all emitters by the same value. Thus, during radar operation within one preset form of expanded beam, different values of beam width can be formed providing prompt adaptability of the PAA RP to the radar environment.

## Forming expanded beams using the FPRP method

The FPRP method is based on the expanded beam forming using partial RPs formed by the pairs of adjacent emitters [6-10]. Adjacent emitters of linear PAA are usually located at a distance d< λ, therefore, partial RPs represent a wide beam with two side lobes. Let us consider the peculiarities of forming of expanded sector beam using such partial RPs.

Figure 1 shows a typical example of expanded beam forming as exemplified by PAA consisting of М = 30 emitters located with an increment of d = 0.5λ. The cosine-squared-on-a- pedestal AD with the edge field level equal to 0.25 was used. The beam is expanded approximately by a factor of 9 using the FPRP method. The expansion operation involves (М - 1) = 29 partial RPs. The figure illustrates that beam levels are quite low, the level of the maximum partial beam (from the central emitter pairs) equals to ~ 0.054 in amplitude, that is ~ -25.3 dB. The level of beam with a minimum amplitude (from the outermost pairs) is approximately 4 times less.

**Fig. 1**. Forming of expanded beams using the FPRP method: a - phase distribution in PAA aperture; b - amplitude RPs of PAA and partials; c - RPs of PAA and partials in decibels; d - amplitude and phase RP of the 20th partial

In accordance with the expansion algorithm, each partial beam moves to the preset direction (Fig. 1b, c). At a partial beam deviation, its maximum shifts, and the slope of RP phase term changes too (Fig. 1d). As a result of vector superposition of a set of partial RPs, an expanded beam is formed. While, the major part of power of partial RPs is evenly distributed in the angular expansion region, and the minor part is outside this region. In the expanded beam region, the fields of partial RPs are combined vectorially, so that the resulting amplitude increases and, beyond the beam, the fields are compensated. The analysis of provided data allows to understand the degree of precision of the selected control parameters for a set of partial RPs in order to form an expanded beam.

It should be noted that a similar beam expansion option can also be obtained using the conventional method of geometrical optics (GO) [4, 5]. In most of the cases of phase synthesis, the results obtained using the GO and FPRP methods match. However, the use of partial RPs in the FPRP method enables greater flexibility of the synthesis and ensures visual clarity of the synthesis operation itself.

## Scaling properties of expanded beams

The property of scaling involves particular ratios between the phase distribution and beam width [9, 10]. If the phase synthesis solution possesses the property of scaling, the operation of additional expansion of PAA beams can be performed based on the previously found expansion option. Thus, if a family of expanded beams of the same type is to be implemented in a PAA, it suffices to find one primary PD using the FPRP method to expand the beam by a factor of a, and to get all other beams with expansion in αβ through multiplying the primary PD by the β coefficient.

When performing the scaling operation, the following critical features must be taken into account:

- the scaling operation generally requires the use of the primary PD not adjusted to the interval of 2;
- during the operation of scaling by a factor of β, the boundaries of beam start and end change by a factor of β in variable u = sin(θ);
- during scaling of the beam by a factor of β, which was initially expanded by a factor of a using the FPRP method, the PD and beam shape are formed identical to those obtained during the initial beam expansion by a factor of αβ using the FPRP method.

The following circumstances should be also taken into account:

- the scaling coefficient is capped, just like the beam expansion coefficient. The boundaries of an expanded beam cannot exceed the dimensions of the visibility area;
- the FPRP method and scaling property are described as applied to the directivity multiplier (DM) of a multi-element PAA, i. e. for the case of use of omni-directional independent emitters. For real PAAs, the specified theory holds true in case of the synthesis and scaling of beams in the angular region, where the RP of one emitter (scanning pattern) is close to unity. In synthesis of wider beams, the required beam form should be adjusted subject to the scanning pattern;
- in a PAA with symmetrical AD, during formation of sector beams, expanded using the FPRP method, in the sum channel, the RP with extended angular distance between the maxima and monotonic level change is formed in the difference channel.

## Characteristics of the difference RP during expansion of the sum RP beam

Let us consider the characteristics of expanded sector beams as exemplified by the PAA with circular aperture, which consists of ~800 emitters (Fig. 2). The figure shows the examples of PD (Fig. 2b) and expanded sector beams by the sum channel (Fig. 2c), obtained using the method of scaling relative to the beam initially expanded by a factor of 3 (PD and RP are green), and a series of relevant RPs by the difference channel (Fig. 2d). The scaling was performed with the same coefficients β = 2; therefore, RP beams by the sum channel are expanded by a factor of 3, 6, 12, 24 in variable u = sin(θ).

In RP calculation, the pattern of one emitter within the array of type F1(θ) = cos(θ) was used. It should be noted that the PD and beam form are only scaled by the sum channel. In general, the shape of the difference RP and its PD have no scaling property in its true sense, except for special cases of an odd number of scaling coefficients β=(2n + 1). Moreover, in these cases, one can say that there is just an expanded angular range of the initial part of the difference RP. The above data suggests that this range is expanded more slowly than the beam of the sum RP; besides, during the expansion, the form of ascent part is changed and levels of side lobes increase substantially.

The form distortion factor of initial part of the difference RP is of great practical importance. To estimate distortions of the above-considered beam expansion options, approximations of initial parts of the difference RPs were performed using analytical functions with error estimates. Research has shown that the shape of initial parts of the difference RPs has well-defined regularities of behaviour during scaling of a sum RP beam. In such case, regularities are the simplest when using values of variable u = sin(θ) in X direction. It appears quite explicable, as mathematical expression of the RP contains parameter u in an explicit form. The results of mathematical modelling are shown in Figure 3 and in Table 1.

Table 1

Parameters of R-squared values R^{2}

Кр |
N = 1 |
N = 2 |
N = 3 |
N = 4 |

1 |
0.9502 |
0.9994 |
1 |
1 |

3 |
0.9702 |
0.9998 |
0.9998 |
1 |

6 |
0.995 |
0.9954 |
0.9961 |
0.9999 |

12 |
0.9608 |
0.9929 |
0.9975 |
0.9978 |

24 |
0.9796 |
0.9969 |
0.999 |
0.999 |

Figure 3 shows initial parts of the difference RP before its maximum for the beams shown in Figure 2. It turned out that, with a very high precision, these parts are described by polynomials of degrees from N = 2 to N = 4 (Table 1), while the practical application can be limited to degree 1 and 2 polynomials, especially for beams with small expansion and in a limited initial part. The approximation was performed in MS Excel, determination coefficient R^{2} was used as an accuracy factor.

## Generalized parameters of expanded beams

Generalized characteristics calculated within the range of expansion coefficients are of main interest. Their analysis allows to identify general regularities of behaviour of the characteristics important for practical application.

Figure 4 provides generalized characteristics of the RP by sum and difference channel for the selected PAA. One expansion option was calculated, as all other options were found using the scaling method. To estimate the sector beam, own normalised coefficients of beam expansion by levels of -3 dB (Кр_{-3}_{дБ}) and -6 dB (Кр_{-6}_{дБ}) were used, and for the difference RP - normalised angular distance between the first maxima (Кр_{разн}). Therefore, values of all three dependencies of expansion coefficients begin with unity. For the sum RP, the expanded beam power level normalised by the level of the in-phase aperture beam was also monitored. For expansion coefficients, the left vertical axis of the graph was used and for the expanded beam level - the right one.

For the considered PAA, the maximum values of expansion coefficients reach the following values: Кр_{-3}_{дБ} ≈ 28, Кр_{-6}_{дБ} ≈ 22, Кр_{разн} ≈ 7.5. The acceptable form of the difference RP is attributable to the symmetry of the initial AD and PD expanding the sum RP beam.

Analytical approximations were determined for the specifi ed parameters. The resulting R-squared values for all dependences are no worse than R^{2}= 0.99. The dependence of expanded beam level on expansion coefficient is expressed by the formula y = 0.9627(Кр_{–3дБ})^{-0.949} at R^{2}= 0.9982, which is very close to the inverse square law.

It is noteworthy that the beam by level of -6 dB is expanded in strict proportion to its expansion coefficient by level of -3 dB. This indicator characterizes the linear dependence of decrease of the RP level beyond the beam being formed, at least in the initial drop part.

The generalized regularity of behaviour of the angular separation between maxima of the difference RP is critical. The function graph has essential non-linearity. Modelling results suggest that this function with a high degree of accuracy follows the regularity y = 0.996(Кр_{–3дБ})^{0.6174} at R^{2} = 0.993, which is close to the square root law.

It should be noted that the differences between actual and theoretical results are partly explained by the fact that beam expansion algorithms are developed for the PAA radiation multiplier, and RP was calculated using the RP of a single emitter of type F1(θ) = cos(θ). Oscillations on the top of the expanded beam manifested itself too. The differences in parameters of the expanded beams are most notable at larger expansions close to finite expansions.

Note that, although the considered results were obtained for PAA with specific geometrical parameters and excitation law, there is reason to believe that the described effects in the range of expansion coefficients will be similar for PAA with other dimensions and another amplitude distribution in the aperture.

## Conclusion

Features of phase synthesis of beams expanded in one dimension using the FPRP method were considered. It was demonstrated that discovery of the scaling feature of phase solutions makes it possible to substantially expand the area of use of phase synthesis results and to enhance adaptability of multifunctional radars to the radar environment.

The paper provides main properties and peculiarities of the scaling operation. An example was provided to demonstrate the beam expansion operation using 29 partial RPs.

Generalized characteristics of behaviour of sectoral beams were analysed in the range of change of beam expansion coefficients by the sum channel. The utmost attention was paid to the parameters of the difference channel RP. While modelling using the FPRP method, one sum channel beam expansion option was determined, and all other options were found using the method of scaling of the phase distribution of the same channel. Modelling of expanded beams as exemplified by the PAA with circular aperture consisting of ~800 emitters showed that:

- maximum values of the beam expansion coefficients by the sum channel on levels of -3 dB (Кр
_{-Здв}) and -6 dB (Кр_{-бдв}) reach the following values: Кр_{-Здв}≈ 28, Кр_{-бдв}≈ 22; - during beam expansion by the sum channel, RP with expanded angular distance between main maxima and monotonic level change is formed in the difference channel;
- the form of initial part of the difference RPs is described by small-degree polynomials;
- the normalised angular distance between the first maxima by the difference channel reaches the value Кр
_{разн}≈7.5; - the generalized regularity of behaviour of the value of angular separation between maxima of the difference RP with a high degree of accuracy (R
^{2}= 0.993) follows the regularity y = 0.996(Кр_{–3дБ})^{0.6174}, which is close to the square root law.

The obtained results make it possible to arrange prompt, predictable and coordinated parameter control of the sum and difference RP of a PAA.

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### About the Authors

**O. V. Pavlovich**Oleg Vadimovich Pavlovich – Engineer of the 2nd category

Research interests: mathematical modelling of the radiation characteristics of phase antenna arrays and active phase antenna arrays, phase synthesis of beams having a special form, digital chart formation, signal processing.

**A. N. Gribanov**Alexander Nikolaevich Gribanov – Cand. Sci. (Engineering), Sectoral Head

Research interests: mathematical modelling of radiation characteristics of phase antenna arrays and active phase antenna arrays, phase synthesis of beams having a special form, assessment and processing of dynamic radiation patterns.

**S. E. Gavrilova**Svetlana Evgenievna Gavrilova – Engineer of the 1st category

Research interests: mathematical modelling of radiation characteristics of phase antenna arrays and active phase antenna arrays, phase synthesis of beams having a special form, assessment and processing of dynamic radiation patterns.

**G. F. Moseychuk**Georgy Feodosievich Moseychuk – Head of Laboratory

Research interests: active and passive antenna arrays, microwave technologies, control of the shape of radiation patterns and radiation regimes, methods of antenna measurements, including dynamic radiation patterns, tuning of phase antenna arrays and active phase antenna arrays.

**I. A. Kuznetsov**Ivan Alekseevich Kuznetsov – Engineer

Research interests: electrodynamic modelling of antenna devices, radio systems for special and civil purposes.

### Review

#### For citation:

Pavlovich O.V., Gribanov A.N., Gavrilova S.E., Moseychuk G.F., Kuznetsov I.A. Generalized characteristics of expanded sector beams formed by phased antenna arrays obtained by scaling phase distributions. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2020;(1):46-54.
https://doi.org/10.38013/2542-0542-2020-1-46-54