﻿ 恒定束宽换能器（CBT）阵列工作频率上限分析
 舰船科学技术  2018, Vol. 40 Issue (4): 99-104 PDF

Analyzing the upper limit of the operating frequency for the constant beamwidth transducer (CBT) array
FENG Xue-lei, CHEN Nan-ruo, LI Xiao-wei, WEI Ning-yang
State Key Laboratory of Deep-sea Manned Vehicles, China Ship Science Research Center, Wuxi 210482, China
Abstract: Constant beamwidth transducer (CBT) array achieves broadband constant beamwidth using frequency-independent element weightings. Conventional methods usually underestimate the upper limit of the operating frequency for the CBT array. Therefore, the upper limit of the operating frequency for various CBT arrays is analyzed. Firstly, the upper limit of the operating frequency is analyzed theoretically. The results indicate that the upper limit of the operating frequency for the delayed-curved straight-line CBT array can be achieved by array theory, but it is quite complicated in the case of circular CBT array. Secondly, the upper limit of the operating frequency for the circular CBT array is analyzed using numerical method, and the fitting model is also provided. Thirdly, the upper limit of the operating frequency for the two-dimensional CBT array is analyzed, and the effects of mainlobe ripple and sidelobe amplitude are also discussed. The results indicate that the upper limit of the operating frequency is greater than the result achieved by the array theory for the circular, spherical and cylindrical CBT arrays.
Key words: constant beamwidth     upper limit of the operating frequency     Legendre-function shading
0 引　言

1 阵列工作频率上限理论分析 1.1 直线形阵列

 $p\left( \theta \right) = \frac{{{\rm{e}}^{ - {j}{kr}}}}{r}\int_{ - L/2}^{L/2} {A\left( l \right){{\rm{e}}^{{j}kl\sin \theta }}{\rm{d}}l}\text{。}$ (1)

 $p\left( \theta \right) = \frac{{{{\rm{e}}^{ - {{j}}k\left[ {r + \left( {N + 1} \right)d/2} \right]}}}}{r}\sum\limits_{n = 1}^N {A\left( n \right){{\rm{e}}^{{{j}}knd\sin \theta }}}\text{。}$ (2)

 $F\left\{ {A\left( n \right)} \right\} = \sum\limits_{n = 1}^N {A\left( n \right){{\rm{e}}^{{{j}}n\varOmega }}}\text{。}$ (3)

1.2 圆弧形阵列

 $p\left( \theta \right) = \frac{{{{\rm{e}}^{ - {{j}}kr}}}}{r}\int_{ - \pi }^\pi {A\left( \psi \right){{\rm{e}}^{{{j}}k{r_0}G\left( {\varphi ,\theta ,\psi } \right)}}{r_0}{\rm{d}}\psi }\text{。}$ (4)

 $p\left( \theta \right) = \frac{{{{\rm{e}}^{ - {{j}}k\left( {r + a\cos \theta } \right)}}}}{r}\sum\limits_{n = 1}^N {A\left( n \right){{\rm{e}}^{{{j}}k{r_0}\cos \left( {n\Delta \psi - \theta - \frac{{N + 1}}{2}\Delta \psi } \right)}}}\text{。}$ (5)

 图 1 直线形和圆弧形阵列示意图 Fig. 1 Geometries of a straight-line array and a circular array
2 阵列工作频率上限数值分析 2.1 一维阵列

 $A\left( \psi \right) = \left\{ {\begin{array}{*{20}{c}}{{{{P}}_\nu }\left( {\cos \psi } \right),\;\;\;\;\;\;\psi \leqslant {\theta _0}}\text{，}\\{0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\psi > {\theta _0}}\text{。}\end{array}} \right.$ (6)

 $p\left( \theta \right) \approx \left\{ {\begin{array}{*{20}{c}}{\rho c{r_0}{{{P}}_\nu }\left( {\cos \theta } \right)\frac{{{{\rm{e}}^{{{j}}k\left( {r - {r_0}} \right)}}}}{r},\;\;\;\;\;\;\theta \leqslant {\theta _0}}\text{，}\\{0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\theta > {\theta _0}}\text{。}\end{array}} \right.$ (7)

 图 2 圆弧形CBT阵列示意图 Fig. 2 Geometries of the circular CBT arrays

 图 3 不同情况下圆弧形CBT阵列的波束 Fig. 3 Beam pattern of the circular CBT array in various cases

 图 4 圆弧形CBT阵列的工作频率上限 Fig. 4 Upper limit of the operating frequency of the circular CBT array

 $\frac{d}{\lambda } = a\left( {1 + \frac{b}{N}} \right) \cdot \left( {1 + c{\theta _0}} \right)\text{，}$ (8)

 $\frac{d}{\lambda } = \left( {1 - \frac{{2.7}}{N}} \right) \cdot \left( {1 - \frac{{{\theta _0}}}{{1000}}} \right)\text{。}$ (9)

2.2 二维阵列

 $A\left( {x,y,z} \right) = \left\{ {\begin{array}{*{20}{c}}{{{{P}}_\nu }\left( {\frac{z}{{{r_0}}}} \right),\;\;\;\;\;\theta \leqslant {\theta _0}}\text{，}\\{0,\;\;\;\;\;\;\;\;\;\;\;\;\;\theta > {\theta _0}}\text{。}\end{array}} \right.$ (10)

 图 5 二维CBT阵列示意图 Fig. 5 Geometries of the two-dimensional CBT arrays

 图 6 圆弧形CBT阵列的工作频率上限 Fig. 6 Upper limit of the operating frequency of the circular CBT array

 $\frac{d}{\lambda } = \left( {1 - \frac{{2.3}}{N}} \right) \cdot \left( {1 - \frac{{{\theta _0}}}{{400}}} \right)\text{。}$ (11)

3 讨论

 图 7 不同情况下的拟合参数 Fig. 7 Fitting parameters in various cases

 图 8 使90%情况下都满足要求的工作频率上限 Fig. 8 Upper limit of the operating frequency at which 90% of the cases meet the requirements
4 结　语

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