Capacitance matrix values for the structure for the square conductors with a circular shield (see Figure 6).
The research described in this chapter analyses two-dimensional potential problems for the multi-body systems, transverse electromagnetic wave propagation along multi-conductor transmission lines and two-dimensional plane wave scattering by various arrays. All conductors may be of arbitrary cross-sections; the only restriction on the system geometry is a smooth parameterization. These problems are mathematically modelled by Dirichlet boundary value problems for either the Laplace or the Helmholtz equation, with the classical integral representation of the solutions in the form of single-layer potential. The analytical-numerical algorithm presented here is based on the method of analytical regularization. The key idea behind this technique is an analytical transformation of the initial ill-posed integral equations to a well-conditioned Fredholm second kind matrix equation. The resulting system of infinite linear algebraic equations is effectively solved using the truncation method: the solution of the truncated system converges to the solution of the infinite system with the guaranteed accuracy that only depends on the truncation number and thus may be pre-specified. The solution obtained is applied to the accurate analysis of 2-D electrostatic- and electrodynamic-field problems for multi-conductor systems with arbitrary profiled conductors. Examples of some conceptual shielded transmission lines incorporating various configurations of conductors and scattering problems for the arrays of thick strips establish the utility of our method and its reliability in various situations
- analytical regularization
- Laplace equation
- Helmholtz equation
In electrostatic and electromagnetic studies of highly elongated cylinders, and ensembles or arrays of such cylinders, it is well-known  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 , where is the length of the cylinder, and
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  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 , 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 : this paper studies single and double periodic devices using a semi-analytical mode-matching technique. Diffraction of the TM-polarized Gaussian beam by
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 , point-matching method , boundary element method , spectral-space domain method , finite element method [18-20], transverse modal analysis  and mode-matching method . A numerical integral equation approach is used in  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 ). 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 , and in  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 is given by
where is related to the linear charge distribution on the contour in the case of the Laplace equation, and to the linear current density in case of the Helmholtz equation. is the Green’s function for Laplace’s equation in 2D space. If the contour is charged to some prescribed potential value , then may be found by solving the equation:
This equation may be classified as a first kind Fredholm equation with a singular kernel; it is ill-posed . 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 . 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
The problem is to find electrostatic potential elsewhere inside the shielded region. This electrostatic problem is fully described by the Dirichlet boundary value problem for the Laplace equation:
with boundary conditions of the potentials given at the surface of each of
To employ the regularization procedure, all contours must be smooth enough and non-self crossing to provide their continuous parameterization and twice differentiation at each point of .
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 as the sum of the single-layer potentials contributed by each cylinder:
where is the unknown line charge density of the th conductor scaled by , is the boundary contour of the th conductor and points lie in the area between the contours. The Kernel of the integral equation (5) is the 2D free space Green’s function:
where is the distance between points and .
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 and introducing some new notations:
we obtain the system of
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 such that , points corresponding to and belong to different contours and so everywhere; hence, the corresponding integral terms do not contain singularities. For the corresponding integral contains a singularity of logarithmic type at the points . In this case, we analytically separate the Green’s function into the singular part and a remainder that does not contain any singularity:
Now we can determine from (10) as follows:
The function is a regular function, defined everywhere except at points ; the function is defined everywhere. It can be shown that for the Laplace’s equation this regular function has the same degree of smoothness as the contour parameterization. An exact expression for at the points of singularity where was obtained analytically:
where is an arc length in the point .
Now we can redefine function everywhere by the formula:
Using the well-known Fourier expansion, we can formulate an expression for the singular part of the Green’s function:
As the function is regular, we can expand it into double Fourier series:
Also the unknown function and the given potential function are represented by their Fourier series:
After substitution of all expansions into (9), one can arrive at the system of integral equations:
Using orthogonal properties and completeness of the functions and defining the rescaled unknown Fourier coefficients of charge density function as follows: when and we obtain the following infinite system of linear algebraic equations:
Following the steps suggested in , 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 arbitrary profiled perfect electric -conductor cylinders embedded into a homogeneous dielectric medium with relative permittivity . The main steps of the solution algorithm are similar to those which were carried out to obtain the solution to the Laplace equation, presented in Section 2.2.
The scattered electromagnetic field obeys the Helmholtz equation:
where point lies exterior to the structure , is the wave number and is the corresponding wave length.
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 (), therefore the scattered field should satisfy the Dirichlet boundary condition on metallic surfaces:
The field should also satisfy the Sommerfeld radiation condition:
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 is the relevant free space Green's function depending on the distance between the observation point and point lying on the contour and the function is related to the linear current density as (where is a wave number, the light speed).
Applying boundary conditions to (23), we obtain the system of integral equations:
After parameterization of the contours where , we use the definition of a line integral and obtain a functional equation in the form:
The following notation is used in (25): where is the distance between points and lying on the -th and -th contours, respectively.
The kernel of the integral equation (25) contains a singularity only in the terms . It is of logarithmic type at the points , and we analytically split the Green's function into a singular and a regular part similarly to the solution steps for the Laplace equation in Section 2.2:
so that the regular part of Green's function is
The function is a regular function, defined everywhere except at the points ; the function is regular everywhere. An exact expression for at the points of singularity is obtained analytically:
where is Euler's constant.
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 : The unknown function is also represented by its Fourier series:
After substitution of all expansions into (25), one can arrive at the system of integral equations. Following the regularization steps for the Laplace equation from Section 2.3, we obtain an infinite system of linear algebraic equations of the second kind:
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 . 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  and the coaxial line with a shifted inner conductor . Results obtained coincide for up to 16 decimal places with the published solutions starting with for the centred inner conductor and for the inner conductor located close to the shield. Our results are also in a good agreement with other semi-analytical and numerical techniques (for example, presented in ).
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 does not exceed 4.5 s on a standard PC.
Efficiency of the developed method is also illustrated by the behaviour of the normalized truncation error versus truncation number (see ) calculated in the maximum norm sense as:
where denote the solutions to the systems (19) or (29) truncated to equations, respectively. The results of the calculations of truncation error for the infinite system (18) defining the solution to the Laplace equation are shown in the Figure 2. The considered structure is a circular shield of radius 1 with elliptic conductor with major semi-axis
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 . Arc length parameterization is one way to overcome this drawback. Other parameterizations could be even more effective, but they often require some adjustments for each shape.
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 and its derivative are continuous , with
In this equation, and are the figure size parameters, define corner sharpness and represents the symmetry. This formula allows us to model a great variety of shapes such as an ellipse (), a rectangle with rounded-off corners (), a star with the smooth rays (is equal to the number of rays) and many others.
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 (see (32)); the greater the is, the sharper the corners of the rectangle are. Sharper corners require higher truncation number to get the same level of the accuracy due to the parameterization by the angle. In Parameterization 2, is a radius of curvature used to smoothen the corner, is rectangle height. In all cases, rectangle height/width ratio is equal to 0.5. Parameterization 2 is not twice differentiable - there is a discontinuity in the second derivative at the joining of the straight line and the quarter circle. This account for slow convergence of the second parameterization as .
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  and capacitance calculations for the capacitance microscope .
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 in the case of the circular shield and conductors of nearly rectangular cross-section (Figure 4). Here is a total charge on the th cylinder, and is the potential of the th cylinder.
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 is satisfied, where is the distance between the conductors, is a parameterization step, is a maximum contour length.
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  and then studied by different authors with many variations . In addition, periodic structures in the millimetre wave range with high precision requirements must be planar structures, for fabrication reasons . Also, in other applications the strip grating is often used as a circular polarizer . 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 ; 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 .
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 and its thickness . The element spacing is characterized by . The incidence angle is . In our calculations, we set , so that is the relative wave number.
The radar cross-section (RCS) is determined from the scattered field as in the direction , where is angle of incidence of the incident plane wave, via
The dependence of the RCS (, in dB) on the incidence angle, , for the 3-, 5- and 9-elements array () is shown in Figure 9. As the number of elements grows two peaks occurring away from normal incidence begin to dominate over the other minor maxima, due to specular reflection from the elements of the array. The highest peak corresponds to normal incidence (
In our calculations, we fix the number of the strips in the grating and calculate the current distribution on each element. The results are shown in Figure 10, where we used the parameters: .
Because of the symmetrical (for normal wave incidence) location of the elements in the array, the distribution will be identical for the strips, numerated by the indexes and . Figure 10 demonstrates that current density distributions on all strips are very close to each other, as a front planar surface of each strip is uniformly illuminated by the incidental plane wave. This induces, in particular, a constant current density on most part of the frontal surface of the strip, except in the narrow region near its corners. The current density has predictable jumps at the angle values, corresponding to the corners of the thick strips. Figure 10 shows four sharp peaks in the distribution of the function . Two dominant peaks correspond to the directly illuminated corners of the frontal surface of the strip. Two other peaks with significantly smaller magnitude correspond to the corners on the underside of the strip; furthermore, the current density on these parts is relatively small. Due to these peculiarities of the current density distribution on the surface of the thick strip, the dependence of on the number of strips in the array is quite weak.
The scattered pattern in the direction is defined as
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, , and wave number . With a small number of strips in the array, the shape of the main beam is not symmetric (the case , Figure 12); for formation of a well-focused beam, it is necessary to increase the number of strips in the array. This assertion is confirmed by the substantially improved shape of the main scattering beam when element number increases to .
Next we consider the frequency dependence of the RCS on wavenumber. Setting the array parameters: , we calculate the function in the range (see Figure 14).
It is worth noting that extremely high values of the function are explained by its normalization (the single element width was chosen to be a characteristic parameter, so that . For the 9-element array analysed above the total width of the array is units (all sizes are related to the strip width); hence, . For comparison with the case of a single element case, the RCS should be scaled by the total width of array.
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 . At the lower values of the relative wave number (), there is no influence of the non-symmetrical location of the central strip on the value of the RCS. At the value , the shift (see Figure 15) becomes slightly greater than the wave size . Hence, we can conclude that the disturbance in the location of the strip in the array is insignificant on the RCS when the shift does not exceed .
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 greater than its height (); otherwise, it is more reasonable to call such a structure a rectangular or square bar (). The ratio represents a threshold value; the parameters and are better described as height and width as shown in Figure 16.
Here we consider normal wave incidence on the array of strips with the fixed parameters: and . Starting from the relative strip thickness , we consider the dependence of RCS on for a 3-element array in the range . The results of these calculations are presented in Figure 17 (), Figure 18 () and Figure 19 ().
Surprisingly the RCS is fairly insensitive to substantial thickening of the original horizontal thin strip (), even at the extreme transformation of the strip into a vertical rectangular cylinder with . The difference between the maximum value and its minimum value for the wavenumbers and does not exceed . In the case when , this difference increases to .
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 by the characteristic maximum size of the structure. We define geometric parameters for the single reflector and for the array of the corner reflectors (see Figure 20) by , where is the distance between strip centres, and are the single strip relative width and thickness, respectively. We calculate the dependence for the three values . As in previous calculations, we take . The E-polarized plane wave excites the truncated corner reflector with incident angle . The frequency dependence of the RCS in the range for the single truncated corner reflector is shown in Figure 21.
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 ; the average level of the RCS steadily grows along with the relative wave number. Combined as a 3-element array, a moderate enhancement of the RCS is observed. Closer spacing () of the reflectors preserves the regular oscillatory behaviour of the , in contrast to more separated spacing (), where some peaks in the RCS graph (red, Figure 22) become suppressed.
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 . The computation of the matrix elements is based on the discrete FFT, making the matrix filling an accurate and extremely fast procedure. The only limitation imposed on the contour is its smoothness. When the contour incorporates corners, they should be rounded. The developed algorithm is numerically stable and fast, and accuracy of the solution can be pre-specified. The solution obtained is applied to the accurate analysis of 2D electrostatic and electrodynamic field problems for multi-conductor systems with arbitrary profiled conductors. Examples for some conceptual shielded transmission lines incorporating differently profiled conductors and scattering problems for the arrays of thick strips illustrate the efficiency and reliability of the method.
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