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Antenna array

Complex directional antenna consists of separate near-omnidirectional antennas (radiating elements) positioned in the space and driven by high-frequency …

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Viсtor Ivanovich Chulkov, Leading Researcher of Research Institute (Kaluga, Russia).
He is the author and head of the project “EDS–Soft” since 2002.

Application of iterative method of calculation of full electromagnetic field for conformal antenna arrays

Published: 04/19/2007
© V. I. Chulkov, 1991. All rights reserved.
© EDS–Soft, 2007. All rights reserved.

The work [1] describes one of the variants of the boundary layer method, based on the asymptotical solution of the bound system of parabolic equations, directly following the Helmholtz equations. The solution itself is constructed as quadratures (or rows), which is composed of Airy functions on the conditions that the conformal surface with impedance structure (SIS; surface S) has large electric size.

V. V. Koryshev in relation to cylindrical surfaces suggested constructing of Green tensors based on the iterative procedure of solving the boundary problem of electrodynamics for flat SIS. It is suggested that this method is applicable for convex surfaces of arbitrary radius of curvature and . In this article are generalized for double curvature SIS.

Points, located above the S surface can be unambiguously described by the radius vector [2]:


where , — coordinates of points on the S surfaces, specified by the radius vector , and the direction of the axis is specified by the direction of the normal vector to S . The covariant basis above the S surface taking into account the derivative Weingarten formula [3] can be written as follows:


where and – covariant basis on the surface S, , – components of the secondary quadratic form of the surface S, – contravariant components of metric tensor.

We will use the following notation: — covariant components of tensor of the used coordinate system ,, , g — determinant of metric tensor,


where , — metric tensor on the S surface.

Heterogeneous Maxwell equations written for covariant coordinates of vector field (dependency on time is specified as ), have as it is known the following form:


where n, k — "dead" indices, — contravariant Levy−Civita pseudotensor, — covariant components of metric tesnor, g — module of determinant of metric tensor, , — covariant components of vectors of secondary field, , — covariant components of vector of magnetic and electric currents on radiators which are sources of secondary waves, — symbol of covariant derivative. Using simple transformations, the equations (4a) can be represented as follows:


where Rot means rotor operation in the Cartesian coordinate system and , are vectors of electric and magnetic steady volume external “currents” caused by the curvature of the surface of the AA:


In the notation (5) — identity tensor, — tensor, covariant components of which are . If the carrier of exterior currents , is finite, then , when according to the uniqueness theorem [4]. As it follows from (4), the original diffraction problem in the convex body can be substituted by the equivalent diffraction problem on a flat surface with existence of currents bulk–distributed over this surface , . The properties of the space above S remain the same. Taking into account (3), we will write in detailed form:


To simplify the recording, we will use the following notation: , , , , m=1,2,3 — orts of the Cartesian coordinate system. According to (4) it can be written as follows:


where V — volume of the entire space involved (0≤z≤∞, -∞≤x,y≤∞), and tensors , are determined by the equations:




, — “reflection” coefficients for flat surface [5]. The remaining tensors are determined from the equations:

In expressions (8) and (9) the top line in curly brackets corresponds to the case , and the bottom one — to the case ; index p=1 corresponds to the H−wave, and index p=2 — to the E−wave, and

We will introduce the notation:

— field six−vector, — current six−vector, , — linear matrix integral operations, cores of which are equal to:

Then, using (5), the expressions (7) can be represented in the operation form:


The equation (10) will be solved by the iterative method, and in correspondence with (4) and (5)

Let’s examine the first approximation, taking as the zeroth approximation the field of flat array , :


where — volume, taken by currents , . After substituting (11) into (7) and reducing the similar terms, we obtain:

where Green tensors of the conformal surface in the first approximation have the following form:


In the expression (12) , when and , when ,

and index "r" denotes regular parts of the tensor Green functions [6]. Usage of tensors (12) means that we passed from electrodynamic examining of the conformal surface to examining of the flat surface when only exterior currents exist: , . Effect of bulk , is taken into account when deriving the expressions (12).

Let’s write the volume integral in (12) in more detail by substituting into it (8) and (9). By considering two characteristic cases and writing the expressions in general form we receive:

— when


— when


The following notation is used:


When calculating using the formulas (13) and (14) the following integrals appear:


where the following notation is used:


When calculating using the formulas (17) it is inconvenient to use the general expressions (6). To simplify the notation, we will assume that , , in non-umbilical points represent an orthogonal coordinate system in the lines of curvature of the surface S [3]. The main directions and can be determined from the equations:


where the following notation is used:

— surface equation of S, , , , and functions and are determined from the conditions of equality of secondary mixed derivatives (consistency condition). Using the notation , these conditions can be written as follows:

they, taking into account (18) reduce to the two independent linear differential equations in partial derivatives:


with initial conditions . The following notation is used here:

By solving the equations (19), we receive [7]:

where and — are corresponding solutions of the normal differential equations and , which determine the equations of projections of curvature lines of the surface S to the plane : and . Functions and — are arbitrary differentiable functions, satisfying the conditions when .

In umbilical points of the surface S we choose as the main directions two mutually orthogonal directions [3].

Thus, in the curvature lines we receive (, , — main curvatures of the surface S):

— when


— when


When deriving the expressions (20a) the formula (2.325.1) from [8] is taken into account, and — is the analytical continuation into the complex area, sliced along the negative part of the real axis, integral exponential function [9]. The remaining integrals in (17) are equal to zero.

Also of interest are the values of some integrals from (20a) when b=∞ [9]:


When S is a cylindrical surface (, , ), the expressions (20a) are simplified:

The expressions (12)…(14) taking into account (15)…(21) completely solve the problem of determining the tensor Green functions of conformal surface in first approximation and, therefore, the first approximation of the electromagnetic field scattered by this surface.


1. Chulkov V.I. Mathematical modeling of antenna array composed of microstrip radiators above the impedance surface of generalized cylinder.— Deposited manuscript NIIEIR, №3–8904, 1991. (In Russian).
2. Corn G., Corn T. Mathematics reference book for scientists and engineers; Edited by I.G. Ara-manovich.— M.: Nauka, 1968.— 720p. (In Russian).
3. Vekua I. N. Basics of tensor analysis and theory of covariants.— M.: Nauka, 1978.— 296p. (In Russian).
4. Markov G.T., Chaplin A.F. Excitation of electromagnetic waves.— M.: Radio i svyaz, 1983.— 295p. (In Russian).
5. Chulkov V.I. Usage of radiant strips in antenna arrays.— Radiotechnics and Electronics, 1992, 37, № 5, p.834…840. (In Russian).
6. Zhuk N.P., Tretyakov O.A. Green functions of Maxwell equations for flat-layered medium.— Radiotechnics and electronics, 1985, 30, №5, p.869-875. (In Russian).
7. Stepanov V.V. Differential equations course.— M.: GIFML, 1958.— 468p. (In Russian).
8. Gradshtein I.S., Ryzhik I.M. Tables of integrals, sums, rows and products.— M: GIFML, 1962.— 1100p. (In Russian).
9. Handbook on special functions/Edited by. M. Abramowitz and I. Stigan /Translated from English by. V. A. Ditkin and L. N. Karmazina.— M.: Nauka, 1979.— 830 p. (In Russian).

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