A general coordinate formula for designing phased array antennas in cylindrical shape with triangular grid

tion pattern
 Assume that the target is at point M, whose direction is represented by the directive
unit vector −→u = (cosθcosϕ, sinθcosϕ, sin ϕ). When the array is analyzed as in Fig. 1b,
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 Journal of Science and Technology - Le Quy Don Technical University - No. 193 (10-2018)
it is convenient to take the phase reference (ψ1 = 0) at position A1 = (R, 0, 0). The
 th
element n at position An = (xn, yn, zn) is set up initial phase (ψn) and amplitude
(an).
 The path-length difference between the element An and A1 directing to M is [1], and
it is:
 −→−−−→ −→ −−→ −−→
 ∆l = u A1An = u (OAn − OA1) (6)
 Substituting the formula (3) and the proposed formulas (4), (5) to equation (6), ∆l
becomes:
   1  p−1  p−1 
 
 

 ∆l = R cos q − 1 + 2 2 − 2 ∆θ − 1 cos θ cos ϕ
  1  p−1  p−1 
 
 (7)
 +R sin q − 1 + 2 2 − 2 ∆θ sin θ cos ϕ + (p − 1)h sin ϕ
The phase difference (∆ψn) between the two elements An and A1 is:
   1 p − 1 p − 1  
 ∆ψ = k∆l = kR cos q − 1 + − ∆θ − 1 cosθcosϕ
 n 2 2 2
 (8)
  1 p − 1 p − 1 
 +kR sin q − 1 + − ∆θ sinθcosϕ + k(p − 1)h sin ϕ
 2 2 2
 2π
 where k = λ is wave number, and is wavelength. On the other hand, array factor
(AF) of the CATG is [3]:
 P ×Q P ×Q
 +∆ψ )
 X jψn j∆ψn X j(ψn n
 AF (θ, ϕ) = ane e = ane (9)
 n=1 n=1
To steer the main beam of the CATG to a particular point M at which direction is
 −→ 0
represented by the directive unit vector u0 = (cosθ0cosϕ0, sinθ0cosϕ0, sin ϕ0) , all
elements in the CATG must have the same phase at M0, which means:
 ψn(θ0, ϕ0) = −∆ψn(θ0, ϕ0)
   1  p−1  p−1 
 
 

 = −kR cos q − 1 + 2 2 − 2 ∆θ − 1 cos θ0 cos ϕ0
  1  p−1  p−1 
 

 −kR sin q − 1 + 2 2 − 2 ∆θ sin θ0 cos ϕ0 − k(p − 1)h sin ϕ0
 (10)
 Replacing formulas (10) and (8) into (9), array factor of the CATG is obtained as
below:
   jkR cos q − 1 + 1  p−1 −  p−1 
 ∆θ
 − 1
  
 exp 2 2 2
 P ×Q  × (cos θ cos ϕ − cos θ0 cos ϕ0) 
 X    1  p−1  p−1 
 
  
 AF (θ, ϕ) = an  jkR sin q − 1 + − ∆θ 
  × exp 2 2 2 
 n=1  × (sin θ cos ϕ − sin θ0 cos ϕ0) 
 × exp (jk(p − 1)h (sin ϕ − sin ϕ0))
 (11)
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Section on Information and Communication Technology (ICT) - No. 12 (10-2018)
 Equation (11) is convenient in order to determine phase distribution and plot radiation
pattern when steering beam at specified direction. In order to take into account the
mutual coupling between elements in the array, the P × Q – element array may be
seen as a P × Q – port characterized by a P × Q scattering matrix with the scattering
coefficients whose direction is represented [3]:
 − 
 Vn 
 Snm = +  for n, m, l = 1, 2, . . .,P × Q. (12)
 Vm  +
 Vl =0,l6=m
 + −
where Vn and Vn are the wave amplitude of, respectively, the incident and reflected
waves at the element nth. When steering the beam to the particular point M0, the
incident wave is applied a phase ψn(θ0, ϕ0)that satisfies equation (10). With the proposed
solution, the scan reflection coefficient of the element nth is determined:
 P ×Q
 P +
 − SnmVm
 Vn m=1
 Γn(θ0, ϕ0) = + = +
 Vn Vn
    1  pm−1  pm−1 
 
 
 
 P ×Q −jkR cos qm − 1 + 2 2 − 2 ∆θ − 1 cos θ0 cos ϕ0
 P  1  pm−1  pm−1 
 

 = Snm exp  −jkR sin qm − 1 + 2 2 − 2 ∆θ 
 n=1
 × sin θ0 cos ϕ0 − jk(pm − 1)h sin ϕ0
    1  p−1  p−1 
 
 
 

 × exp jkR cos q − 1 + 2 2 − 2 ∆θ − 1 cos θ0 cos ϕ0
   1  p−1  p−1 
 
 

 × exp jkR sin q − 1 + 2 2 − 2 ∆θ sin θ0 cos ϕ0 + k(p − 1)h sin ϕ0
 (13)
where qm and pm satisfy the condition of m = Q(pm − 1) + qm . Thus, when taking
 +
the mutual coupling effects into account and representing the incident wave by Vn =
an exp(jψn(θ0, ϕ0)), equation (11) becomes:
 AF (θ, ϕ) =
   jkR cos q − 1 + 1  p−1 −  p−1 
 ∆θ
 − 1
  
 exp 2 2 2
 P ×Q  × (cos θ cos ϕ − cos θ0 cos ϕ0) 
 P    1  p−1  p−1 
 
  
 an(1 + Γn(θ0, ϕ0)) ×  jkR sin q − 1 + 2 2 − 2 ∆θ 
 n=1  × exp 
  × (sin θ cos ϕ − sin θ0 cos ϕ0) 
 × exp (jk(p − 1)h (sin ϕ − sin ϕ0))
 (14)
 Equations (13) and (14) give mathematical model for designers to determine radiation
pattern when mutual coupling between elements in the CATG is considered.
4. Simulation results
 We consider an example of the CATG applied in a sonar system, which has 24
elements (N = 24) on a circle and 16 circles (P = 16) in total. If the active sector is
chosen as one third of the circle, the number of active elements on a circle is 8 (Q = 8),
and the total number of active elements in the array is 8×16 = 128 elements. The angle
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 Journal of Science and Technology - Le Quy Don Technical University - No. 193 (10-2018)
 (a)
 (b)
 Fig. 2. Comparison of array factor a) In azimuth plane (ϕ = 0◦); b) In elevation plane (θ = 60◦)
between two adjacent elements is ∆θ = 15◦ and the angle blocked by the 8 elements
 ◦
is determined θactive = 105 . The simulation is implemented at frequency f = 30 kHz,
while the sound speed in sea water is 1500 m/s, then the distance between two adjacent
 1 ◦
circles is chosen h = 1+sin 45◦ λ ≈ 29.29(mm) (steering beam up to 45 [3]). The radius
of a circle is chosen as R = 5λ = 250(mm) [1], and the desired steering angles in
 69
Section on Information and Communication Technology (ICT) - No. 12 (10-2018)
 Fig. 3. Phase distribution generated by formula (10)
 ◦ ◦
azimuth and elevation planes are θ0 = 60 and ϕ0 = 0 respectively. We assume that
amplitude distribution in the array is equal, each element has the isotropic radiation
pattern, and there is no mutual coupling between them in the array.
 The results of the proposed method and the tool Sensor Array Analyzer of MATLAB
 ◦
are shown in Fig. 2. The cut angle in elevation plane is ϕ0 = 0 (Fig. 2a), and the cut
 ◦
angle in azimuth plane is θ0 = 60 (Fig. 2b). In each figure, there are a dashed curve
resulting from the proposed method and a solid curve resulting from the tool Sensor
Array Analyzer of MATLAB for comparison. It is easy to confirm in these figures an
excellent agreement between the dashed and solid curves, which means that the array
factor radiation pattern generated by formula (11) from our proposal is definitely similar
to that raised automatically by the tool Sensor Array Analyzer of MATLAB. Therefore,
the reliability of our proposal is validated totally, and moreover, in our method, the
desired phase of each element in the array is simply derived by using formula (10) as
demonstrated in Fig. 3, while the tool Sensor Array Analyzer of MATLAB requires users
to enter coordinates of all elements in the array and does not provide the users with
any information of phase distribution.
 In order to obtain simulation results when accounting mutual coupling, equations (13)
and (14) are used. Determination of the scattering coefficients in equation (13) may be
implemented by measurements [1, 3] or mathematical formulas. Consider the CATG
with transducer elements used as references [8, 9], which consist of short cylindrical
transducers distributed in surface of the array. We assume that every element is match-
connected to a voltage source (Snn = 0) [3]. In order to calculate reflected wave
amplitude, scattering cross-section known in radar as the radar cross-section (RCS) is
 λ
used [5]. When the diameter of each transducer is chosen as D0 = 3 ≈ 16.67(mm),
 D0
length of this cylinder is chosen as H0 = 10 ≈ 1.7(mm), the maximum value of RCS
 2
 πD0H0
of each element is σ = λ [6]. With the radar cross-section model, the maximum
 p σ
absolute values of the scattering coefficients are estimated by |S | = 2 .
 nm max 4πr nm
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 Journal of Science and Technology - Le Quy Don Technical University - No. 193 (10-2018)
 (a)
 (b)
 Fig. 4. Array factor when taking mutual coupling into account a) In azimuth plane (ϕ = 0◦);
 b) In elevation plane (θ = 60◦)
Where rnm is distance between the element nth and the element mth of the array.
Because the phase values of the scattering coefficients are proportional the distances,
the complex value of the scattering coefficients are estimated by
 Snm = |Snm| exp(−jkrnm) (15)
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Section on Information and Communication Technology (ICT) - No. 12 (10-2018)
With using estimations of the scattering coefficients determined by the maximum ab-
solute values, the scan reflection coefficients and radiation pattern are calculated by
equations (13) and (14).
 The results of two cases of with and without mutual coupling are shown in Fig. 4.
The cut angle in elevation plane is ϕ = 0◦ (Fig. 4a), and the cut angle in azimuth
plane is θ = 60◦ (Fig. 4b). In each figure, there are a solid curve for the case without
mutual coupling and a dashed curve for the case with mutual coupling.
 It could be seen that in this example, mutual couplings between the elements have
the major effects on the side lobes, but the direction and the beamwidth of the main
beam remain relatively stable. The simulation results for the CATG are the same with
the released results for some cylindrical arrays with rectangular grid in the reference
[1]. Therefore, with a given phase distribution, properties of the main beam will not be
affected too much, that is to say, formula (10) can also be used in the case of taking
mutual couplings into account in this specific case, and the availability of formulas
(4)-(5) is still maintained. With other cases in which mutual couplings yield in other
complex forms, it is required to consider carefully the mathematical formation of mutual
couplings in each case.
5. Conclusion
 In this study, with a simple proposal of coordinate combination, determination of
radiation pattern and phase distribution in cylindrical arrays with triangular grid (CATG)
becomes much more mathematically explicit. The validity and merits of our proposal
are verified by comparing the obtained results with those raised by the tool Sensor
Array Analyzer of MATLAB. The simulation results also indicate that in a specific case
of considering mutual couplings between elements in the array, the proposed formulas
is still available for deriving the properties of the main beam.
References
[1] Lars Josefsson and Patrik Persson (2006). Conformal Array Antenna Theory and Design, John Wiley & Sons.
[2] John L. Butler Charles H. Sherman (2016). Transducers and Arrays for Underwater Sound, Springer International
 Publishing.
[3] Hubregt J. Visser (2005). Array and Phased Array Antenna Basics, John Wiley & Sons.
[4] Yu. B. Nechaev, I. W. Peshkov, and N. A. Fortunova (2017). Cylindrical antenna array development and
 measurements for DOA-estimation applications, Conference on Antenna Theory and Techniques (ICATT), pp.
 155-158.
[5] A. D. Waite (2002). Sonar for Practising Engineers, John Wiley & Sons.
[6] Bassem R. Mahafza, Atef Z. Elsherbeni (2004). MATLAB Simulations for Radar Systems Design.
[7] Matworks, Sensor Array Analyzer Tool, MATLAB, [Online]. Available: 
[8] https://www.thalesgroup.com/sites/default/files/database/d7/medias/documents/thales%20UMS%204110.pdf.
[9] 
 Manuscript received 13-06-2018; Accepted 16-10-2018.
 
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Journal of Science and Technology - Le Quy Don Technical University - No. 193 (10-2018)
 Nguyen Dinh Tinh was born in Thanh Hoa, Vietnam. He received the B.E. (Bachelor of
 Engineering) degree in Electronics-Telecommunications, the M.E. (Master of Engineering)
 degree in Radar Navigation Engineering both from Le Quy Don Technical University, Hanoi,
 Vietnam, in 2008 and 2012, respectively. He is working as a lecturer, at Le Quy Don Tech-
 nical University. His research interests include antenna and microwave circuit design, signal
 processing and sonar engineering.
 Nguyen Tuan Hung was born in Nam Dinh, Vietnam, on August 2, 1985. He received the
 B.S., M.S., and PhD. degrees in the Department of Electrical and Electronic Engineering from
 National Defense Academy of Japan in 2010, 2012 and 2015, respectively. He is now a lecturer
 in Le Quy Don Technical University, Hanoi, Vietnam. His research interests include mobile
 handset antenna miniaturization, antenna and microwave circuit optimal design.
 Trinh Dang Khanh was born in Thanh Hoa, Vietnam. He received the B.E. (Bachelor of
 Engineering) degree in Electronics-Telecommunications, the PhD degree in Electronic Engi-
 neering both from Le Quy Don Technical University, Hanoi, Vietnam, in 1983 and 2000,
 respectively. He has been working as an Associate Professor since 2011 at Le Quy Don
 Technical University. His research interests include microwave circuit design, signal processing,
 radar target recognition and sonar engineering.
 Nguyen Manh Cuong was born in Thai Binh, Vietnam. He received the B.E. (Bachelor of
 Engineering), M.E. (Master of Engineering) degrees both in Electronics-Telecommunications,
 and the PhD degree in Radar Navigation Engineering from Le Quy Don Technical University,
 Hanoi, Vietnam, in 1983 and 2000, respectively. He is working as a lecturer, at Le Quy Don
 Technical University. His research interests include multistatic radar, signal processing, radar
 target recognition, and sonar engineering.
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Section on Information and Communication Technology (ICT) - No. 12 (10-2018)
 Duong Huy Binh was born in Ha Noi, Vietnam. He received the B.E. (Bachelor of Engineer-
 ing) degree in Design and Technology of Radio-Electronic Devices from Bauman Moscow
 State Technical University, Moscow, Russia, in 2011, the M.E. (Master of Engineering) degree
 in Electrical Techniques, Electronics and Telecommunications from The Academy of Cryp-
 tography Techniques, Hanoi, Vietnam, in 2017. He is working as an assistant, at Vietnam
 Government Information Security Commission. His research interests include VLSI design,
 signal processing and embedded systems.
 CÔNG THỨC TỌA ĐỘ TỔNG QUÁT CHO THIẾT KẾ
 CÁC ANTEN MẠNG PHA HÌNH TRỤ
 VỚI CÁC PHẦN TỬ ĐAN CHÉO
 Tóm tắt
 Bài báo đề xuất một mô hình toán học hợp nhất tọa độ trong anten mạng pha hình trụ
 với các phần tử đan chéo, một trong những dạng anten được sử dụng trong các ứng dụng
 sonar. Với giải pháp đề xuất, phân bố pha trong mạng anten này có thể được xác định một
 cách nhanh chóng để tính toán giản đồ hướng khi muốn điều khiển cánh sóng đến hướng bất
 kỳ. Các kết quả được khảo sát trong cả hai trường hợp có và không có ảnh hưởng tương hỗ
 giữa các phần tử trong mạng. Độ tin cậy của phương pháp được kiểm chứng bằng cách so
 sánh các giản đồ hướng thu được từ mô hình toán học đề xuất với các giản đồ hướng được
 tạo ra bằng cách sử dụng công cụ Sensor Array Analyzer của Matlab. Tuy nhiên, công cụ
 của Matlab không cung cấp thông tin về phân bố pha của các phần tử, công thức xác định
 giản đồ hướng và yêu cầu người sử dụng phải nhập tất cả các tọa độ của từng phần tử trong
 mạng. Các công thức phân bố pha và giản đồ hướng được đưa ra trong bài báo là cơ sơ để
 thiết kế anten mạng pha hình trụ với các phần tử đan chéo.
74