1. Introduction
In electrostatic and electromagnetic studies of highly elongated cylinders, and ensembles or arrays of such cylinders, it is well-known [1] that the most important effects can be treated by replacing the three-dimensional structure by the corresponding cross-sectional two-dimensional (2D) profile. Such cylinders are described by
The long-standing interest in the investigation of the electrostatic field in periodic structures continues because of numerous applications. One example is the analysis of the propagation of the transverse electromagnetic (TEM) wave in open and shielded multi-conductor transmission lines [3, 4]. When the contour of a conductor coincides with the coordinate surface of one of the coordinate systems in which the Laplace’s equation is separable, the Fourier method (method of separation of variables) is used. More generally, a variety of potential problems have been solved by the conformal mapping method. These results are described in many classical handbooks and monographs. The number of such solved problems is highly restricted. Nowadays, the need for simulation of devices used in practice requires development of more universal methods to tackle problems with objects of various finite-width shapes. One of such numerous examples is the capacitance calculation for thick electrodes [5] where a physically reasonable meaning of ‘edge capacitance’ arises only because an accurate charge distribution over the whole electrode could not be accurately calculated. Though solutions obtained for single objects may adequately describe the real situation, most practical problems deal with a finite number of objects (say, conductors). Even when a conductor is of canonical shape (circular or elliptic cylinder), the solution of an electrostatic multi-conductor problem for an assembly of cylinders of different radii is a very bulky and lengthy procedure. Solving this problem as a classical boundary value problem for Laplace’s equation and enforcing the pre-assigned boundary conditions at the surface of each conductor, it is necessary to make multiple re-expansions of the eigen functions of the Laplace’s equation in each local coordinate system in terms of that chosen to satisfy the boundary conditions.
Electromagnetic and acoustic problems described by the Helmholtz equation can also be considered in two dimensions. A great variety of publications consider the problem involving infinite gratings. They are often used in antenna applications as polarizers and filters. In [6], a vector diffraction formulation for the analysis of perfectly conducting gratings of finite width and thickness is presented. The grating is assumed to have a finite number of infinitely long arbitrarily shaped rods, and is illuminated by an arbitrary plane wave. Electric and magnetic field integral equations are used to numerically solve the corresponding TM and TE electromagnetic problems. Periodic structures in the millimetre wave range are considered in [7]: this paper studies single and double periodic devices using a semi-analytical mode-matching technique. Diffraction of the TM-polarized Gaussian beam by N equally spaced slits (finite grating in a planar perfectly conducting thick screen) is investigated in [8]. Numerous publications consider the different types of arrays. A general approach was presented in [9] for solving the 2D scattering of a plane wave by an arbitrary configuration of perfectly conducting circular cylinders in front of a plane surface with general reflection properties. Acoustic scattering by a cluster of small sound-soft obstacles was considered in [10]. The 2D scattering of a Gaussian beam by a periodic array of circular cylinders is studied in [11]. A study of the electromagnetic scattering from multi-layered periodic arrays of parallel circular cylinders is presented in [12]. The electromagnetic scattering by multiple perfectly conducting arbitrary polygonal cylinders is analysed in [13].
It should be noted that the long history of solving the Laplace and Helmholtz equations is marked by the development of many numerical methods which are useful in simulation of practical devices. Such methods include the finite difference technique, extrapolation [14], point-matching method [15], boundary element method [16], spectral-space domain method [17], finite element method [18-20], transverse modal analysis [21] and mode-matching method [22]. A numerical integral equation approach is used in [23] to explore plane-wave scattering from a nonplanar surface with a sinusoidal height profile for the case of the magnetic field parallel to the surface ridges (TM polarization). In spite of effectiveness of these methods in many cases and flexibility in geometrical representation of the structures, modelling of ridges still have some substantial drawbacks. Most of the methods require large resources in terms of computational time and storage. Often the solutions obtained with such purely numerical methods need to be verified through comparison to other results: accuracy generally cannot be guaranteed for a greater number of iterations or larger-scale computations. This problem becomes pronounced in some topologically complex configurations. In electromagnetics, the corresponding class of numerical solutions is applicable in the low to intermediate frequency range. Resonant systems behaviour cannot be reliably analysed with the numerical techniques (see [24]). Analytical-numerical methods such as those based on the method of analytical regularization (MAR) are designed to overcome these drawbacks in the resonant regime. A comparative analysis of the MAR and other methods was conducted in [25], and in [26] the distinctive features of each of the discussed methods are clearly described. The above-mentioned methods are mostly suited for analysis of a single or very few conductors. In the case of a significant number of conductors with individual profiles, the effectiveness of such purely numerical methods is highly problematic because of the rapidly growing scale of computations.
In order to address these difficulties, we present here a semi-analytical approach to the analysis of 2D electrostatic and electrodynamic field problems for multi-conductor systems. The problems to be solved are treated as the classical Dirichlet boundary value problems for the Laplace and Helmholtz equations. It is well-known [27, 28] that solutions to the Laplace and Helmholtz equations can be represented as a single-layer potential at points exterior to the body of a single conductor with contour
where
This equation may be classified as a first kind Fredholm equation with a singular kernel; it is ill-posed [29]. Nevertheless, this problem has been tackled by many authors who used direct numerical schemes for solving its discrete analogue in a form of a first kind algebraic equation. Theoretically, any numerical method applied to solve this equation is unable to guarantee uniform convergence, or pre-determined computational accuracy.
The only way to avoid these shortcomings is to transform the initial equation into a second kind Fredholm equation, discretization of which guarantees uniform convergence and any pre-determined accuracy of the numerical solution depending on truncation number. We employ the MAR, in particular, described in [30, 31]. An accurate solution to wave scattering by a single infinitely long cylinder of arbitrary cross-section by the MAR was obtained in [32]. The details of the algorithm for cylinders of closed arbitrary profile are presented in [26, 31]. In this chapter, we generalize the MAR for a multi-conductor potential problem where each body is an arbitrary profiled cylinder.
2. Regularization of the electrostatic problem: MAR
2.1. Problem statement
Consider (N-1) arbitrary profiled charged perfect electric conductor (PEC) cylinders embedded into a homogeneous dielectric medium with relative permittivity
The problem is to find electrostatic potential
with boundary conditions of the potentials
To employ the regularization procedure, all contours
2.2. Problem solution
The main challenge of this problem is that all the conductors are arbitrary-shaped and the classical separation of variables method is not applicable here. We use a more general approach based on an integral representation. Using the superposition principle, we seek the solution for the total field potential
where
where
Applying boundary conditions to (5), one can arrive at the coupled system of integral equations for the unknowns
Equation (7) represents a system of first kind Fredholm integral equations that is generally ill-posed.
The contours of the conductors’ cross-sections should be smooth. Thus for the analysis of the rectangular and square conductors, corners should be smoothed. The two most common types of parameterization are by angle and by arc length. Here, we use parameterization by angle. After parameterization of the contours
(8) |
we obtain the system of N integral equations:
The described approach permits us to consider a broader set of possible boundary conditions than simply a constant, though in the application to be described, a constant is deployed on the RHS of (9).
For the kernels
(10) |
Now we can determine
The function
where
Now we can redefine function
Using the well-known Fourier expansion, we can formulate an expression for the singular part of the Green’s function:
As the function
Also the unknown function
After substitution of all expansions into (9), one can arrive at the system of
Using orthogonal properties and completeness of the functions
Following the steps suggested in [33], it can be shown that coefficient matrix in (18) is square summable:
Thus the infinite system (18) is of a second Fredholm kind and can now be effectively solved by a truncation method. The solution of the truncated system monotonically and rapidly converges to the exact solution. The above solution automatically incorporates the reciprocal influence of all charged cylinders, allowing accurate calculation of the line charge densities on the boundaries and then the field potentials at any point of the space between the conductors. Fourier expansions in (18) are calculated numerically as all functions are regular.
3. Regularization of the Dirichlet problem for Helmholtz equation: MAR technique for the N-body multiple scattering
3.1. Statement of the problem
In this section, we consider the scattering problem for the structure which consists of
The scattered electromagnetic field
where point
Here we consider incident fields in the form of a plane wave. We focus on a transverse magnetic (TM) wave polarization of the incident field (
The field should also satisfy the Sommerfeld radiation condition:
As |p|→∞ where
3.2. Regularized solution
Solutions to the Laplace equation can be represented as a single-layer potential at points exterior to the body. Using the superposition principle, we seek the solution as the sum of single-layer potentials contributed by each cylinder:
Here
Applying boundary conditions to (23), we obtain the system of
After parameterization of the contours
The following notation is used in (25):
The kernel of the integral equation (25) contains a singularity only in the terms
so that the regular part of Green's function is
The function
where
We expand the singular part of Green's functions in the same way as in Section 2.2, and perform the double Fourier series expansion for the regular function
After substitution of all expansions into (25), one can arrive at the system of
where the following notations are used:
The infinite systems (31) can be effectively solved by a truncation method. The solution of the truncated system steadily and rapidly converges to the exact solution [34]. There are no limitations on the number of cylinders with arbitrary smooth cross-sections.
4. Numerical results
The numerical code was validated by comparing obtained results with known analytical solutions for the coaxial line with a centred inner conductor [35] and the coaxial line with a shifted inner conductor [36]. Results obtained coincide for up to 16 decimal places with the published solutions starting with
As an illustration of the effectiveness of the obtained solution, we calculate the capacitance matrix for the assembly of arbitrary profiled cylinders located inside the grounded shield. There are no limitations on the number of cylinders with arbitrary smooth cross-sections. The high efficiency of the code is also the result of employment of the discrete Fast Fourier Transform. This makes filling of the matrix very fast routine procedure. For example, the computation time for a problem with the four inner cylinders and truncation number
Efficiency of the developed method is also illustrated by the behaviour of the normalized truncation error versus truncation number (see [30]) calculated in the maximum norm sense as:
where
Figure 3 shows the condition number behaviour in the same case. The results are quite accurate and stable: for a simple structure like this, the condition number has reached a stable value even for small values of the truncation number.
In these examples, the ellipse is parameterized by the angle as a parameter. Fewer number of points on the sides of a slender ellipse results in decreasing accuracy for smaller
Various shapes of conductor will be considered in this chapter. For all system configurations here and below, the inner conductors’ potentials are set to be 1; the shield is grounded. The profile of each interior conductor is described by the super-ellipse equation (32), where function
In this equation,
This parameterization is infinitely differentiable which gives us a great advantage in accuracy. To demonstrate this property, comparison of two different parameterizations used in the solution to the Helmholtz equation for a single rectangle with rounded-off corners is presented in Figure 4.
Parameterization 1 stands for a super-ellipse formula; straight lines with a combination of quarter circles are used for the Parameterization 2. The super-ellipse parameterization uses
4.1. Electrostatic problems
4.1.1. Multi-conductor transmission lines
Here the power of the method is illustrated by the analysis of multi-conductor transmission lines. Other possible applications of our method for the problems modelled by the Laplace equation include impedance calculations for the transmission lines with adjustable inner conductor, published in [39] and capacitance calculations for the capacitance microscope [40].
The distribution of the electrostatic field for a conceptual configuration of a shielded three-conductor transmission line is shown in Figure 5.
It is worth noting that apparently sharp edges are in fact not sharp but have a very small radius of curvature at some points due to parameterization of the contours.
Next we present the calculations for the capacitance matrix (Table 1) calculated by formula
Symmetric alignment: | Non-symmetric alignment: |
| |
Two configurations are examined - a symmetrical one and another obtained by the translation of one conductor, as indicated in Figure 6. In each case, accuracy was ensured by examining error estimates as a function of truncation number as explained above.
4.1.2. Transmission lines with the closely spaced conductors
Another example demonstrating the effectiveness of the developed algorithm is a study of the closely spaced conductors case (Figure 7). In Table 2, capacitance values for a circular conductor are shown depending on a distance between the inner conductor of a radius 0.1 and a shield of radius 1. Truncation numbers are chosen to ensure capacitance values and are stable to four decimal places.
The analysis shows that reliable results are obtained when the condition
4.2. Scattering of a plane wave by an array of thick strips
Arrays, which are composed of a finite number of strips, are probably the most common periodic structures. They are employed in various electromagnetic radiating and wave-guiding devices. For example, a simple but effective leaky-wave antenna can be designed by placing a microstrip grating above a ground plane, as first proposed by Honey in the 1950s [41] and then studied by different authors with many variations [42]. In addition, periodic structures in the millimetre wave range with high precision requirements must be planar structures, for fabrication reasons [7]. Also, in other applications the strip grating is often used as a circular polarizer [43]. The list of applications can be continued. Nowadays, a lot of attention is paid to more realistic models of the strip gratings: finite gratings, excited by compact directional sources [8]; gratings with thick strips [44, 45]; special elemental positioning [46, 47], etc.
In this section, we consider scattering of an obliquely incident E-polarized plane wave by a finite array of metallic thick strips which is relevant to the problems examined in the papers mentioned above. The case of the array of circular cylinders, including resonant effects, was considered in [48].
4.2.1. Linear array of horizontal thick strips
We consider the scattering of the E-polarized plane wave obliquely incident the linear array composed of the metallic thick strips, as shown in Figure 8. The elemental thick strip is described by its width
The radar cross-section (RCS) is determined from the scattered field as
The dependence of the RCS (
In our calculations, we fix the number of the strips
Because of the symmetrical (for normal wave incidence) location of the elements in the array, the distribution
The scattered pattern in the direction
The distribution of the scattered field in the far-field zone is shown in Figures 11-13.
Figure 13 demonstrates the SP for the 3-, 5- and 9-element array with the same geometrical parameters, as in Figure 11, but for a different incidence angle,
Next we consider the frequency dependence
It is worth noting that extremely high values of the function
The next graph in Figure 15 illustrates the effect of perturbing the periodicity of the array on the RCS: the central strip is moved towards to the neighbouring strip by a distance
Now let us investigate how the thickness of the strips in an array impacts the RCS. In fact, we will consider a more general problem. Usually the term thick strip refers to the strip with the width
Here we consider normal wave incidence
Surprisingly the RCS is fairly insensitive to substantial thickening of the original horizontal thin strip (
4.2.2. Inclined arrays of thick strips
It was shown previously in this chapter that different positioning of the strips affects the reflection properties. Let us consider another geometry, arranging the strips as a 2D truncated corner reflector (see Figure 20).
Here we only consider normal wave incidence, so we normalize the function
The frequency dependence of the RCS in the range for the array of three truncated corner reflectors is shown in Figure 22. The RCS for the single truncated corner reflector is of a regular oscillatory character for each parameter
5. Conclusion
In this chapter, a rigorous approach to the numerical analysis of the multi-conductor problems in electrostatics and multiple wave scattering for metallic cylinders is presented. The problems are treated as a classical Dirichlet boundary value problem for the Laplace and Helmholtz equations. All conductors may be of arbitrary cross-sections; the only restriction on the system geometry is a smooth parameterization of the boundaries.
The 2D multi-conductor problems for the Laplace and Helmholtz equations are rigorously solved by the MAR. The problem is transformed to a numerical analysis of an infinite system of linear algebraic equations of the second kind. This explains its fast convergence and guaranteed computational accuracy, depending only upon truncation number