**4.1. Electrostatic problems**

In this equation, *a* and *b* are the figure size parameters, *n*1, *n*2, *n*<sup>3</sup> define corner sharpness and *m* represents the symmetry. This formula allows us to model a great variety of shapes such as an ellipse (*n* =*n*<sup>1</sup> =*n*<sup>2</sup> =*n*<sup>3</sup> =2, *m*=4), a rectangle with rounded-off corners (*n* =*n*<sup>1</sup> =*n*<sup>2</sup> =*n*<sup>3</sup> >2, *m*=4), a star with the smooth rays (*n*<sup>1</sup> =2, *n*<sup>2</sup> =*n*<sup>3</sup> >2, *m* 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

**Figure 4.** Comparison of super-ellipse and smoothed rectangular parameterizations.

convergence of the second parameterization as *Ntr* →*∞*.

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 *n* =*n*<sup>1</sup> =*n*<sup>2</sup> =*n*3, (see (32)); the greater the *n* 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, *r* is a radius of curvature used to smoothen the corner, *h* 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

in Figure 4.

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### *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 threeconductor transmission line is shown in Figure 5.

**Figure 5.** Conceptual circular shielded transmission line with three inner cylinders. (*a* =*b* =0.6, *m*=5, *n*<sup>1</sup> =2, *n*<sup>2</sup> =*n*<sup>3</sup> =13).

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 *Ci*, *<sup>j</sup>* =∂*Qi* / ∂*uj* in the case of the circular shield and conductors of nearly rectangular crosssection (Figure 4). Here *Qi* is a total charge on the *i* th cylinder, and *uj* is the potential of the *j* th cylinder.


**Table 1.** Capacitance matrix values for the structure for the square conductors with a circular shield (see Figure 6).

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.

**Figure 6.** Symmetrical and shifted inner conductors alignment (*a* =*b* =0.1, *m*=4, *n*<sup>1</sup> =*n*<sup>2</sup> =*n*<sup>3</sup> =40).

### *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 con‐ ductor 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 *d* ≥Δ*l* is satisfied, where *d* is the distance between the conductors, Δ*l* = *L* / *Ntr* is a parameterization step, *L* is a maximum contour length.

Rigorous Approach to Analysis of Two-Dimensional Potential Problems, Wave Propagation and Scattering... http://dx.doi.org/10.5772/61287 193

**Figure 7.** A circular shield with a closely spaced inner conductor.

**Symmetric alignment: Non-symmetric alignment:**

**Table 1.** Capacitance matrix values for the structure for the square conductors with a circular shield (see Figure 6).

error estimates as a function of truncation number as explained above.

**Figure 6.** Symmetrical and shifted inner conductors alignment (*a* =*b* =0.1, *m*=4, *n*<sup>1</sup> =*n*<sup>2</sup> =*n*<sup>3</sup> =40).

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 con‐ ductor 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

The analysis shows that reliable results are obtained when the condition *d* ≥Δ*l* is satisfied, where *d* is the distance between the conductors, Δ*l* = *L* / *Ntr* is a parameterization step, *L* is a

*4.1.2. Transmission lines with the closely spaced conductors*

to four decimal places.

maximum contour length.

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

) *<sup>C</sup>* =( 3.5151 -0.2713 -0.4191


)

*<sup>C</sup>* =( 3.5583 -0.4441 -0.4441 -0.4441 3.5580 -0.4440 -0.4441 -0.4440 3.5580

192 Advanced Electromagnetic Waves


**Table 2.** Capacitance values for the structure with the closely spaced conductors.

### **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 waveguiding 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 *w* and its thickness *t*. The element spacing is characterized by *d*. The incidence angle is *φ*0. In our calculations, we set *w* =1, so that *k* ≡*kw* is the relative wave number.

**Figure 8.** Linear array of horizontal strips.

The radar cross-section (RCS) is determined from the scattered field as *ρ* →*∞* in the direction *φ* =*φ*<sup>0</sup> + *π*, where *φ*0 is angle of incidence of the incident plane wave, via

$$RCS = \lim\_{\rho \to \infty} 2\pi\rho \left| \mathcal{U}^\* \left( \rho, \varphi\_0 + \pi \right) \right|^2. \tag{33}$$

The dependence of the RCS (*RCS*(*φ*0), in dB) on the incidence angle, *φ*0, for the 3-, 5- and 9 elements array (*k* =*π*) 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 (*φ*0 = 90°); the second peak corresponds to the incidence angle *φ*0 = cos-1(1/1.5) = 48.19°. The calculation of the current density distribution on the contour of each element is of practical interest, especially for the situations when the results obtained for infinite gratings are used for finite gratings. It is reasonable to assume that the near equal current distribution on all elements of the finite grating is a plausible argument to treat such grating as a fragment of the infinite grating. This idea was used, for example, by Kalhor in [49].

In our calculations, we fix the number of the strips *N* =5 in the grating and calculate the current distribution on each element. The results are shown in Figure 10, where we used the param‐ eters: *φ*<sup>0</sup> =90 , *k* =*π*, *t* =0.1, *d* =1.5.

Because of the symmetrical (for normal wave incidence) location of the elements in the array, the distribution | *J <sup>z</sup>*(*φ*)| will be identical for the strips, numerated by the indexes *n* =±1 and *n* =±2. 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 | *J <sup>z</sup>*(*φ*)|

**Figure 9.** *RCS*(*φ*0), (*k* =*π*, *t* / *w* =0.1, *d* =1.5).

*4.2.1. Linear array of horizontal thick strips*

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**Figure 8.** Linear array of horizontal strips.

eters: *φ*<sup>0</sup> =90 , *k* =*π*, *t* =0.1, *d* =1.5.

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 *w* and its thickness *t*. The element spacing is characterized by *d*. The incidence angle is *φ*0. In our calculations, we set *w* =1, so that *k* ≡*kw* is the relative wave number.

The radar cross-section (RCS) is determined from the scattered field as *ρ* →*∞* in the direction

 r j p

The dependence of the RCS (*RCS*(*φ*0), in dB) on the incidence angle, *φ*0, for the 3-, 5- and 9 elements array (*k* =*π*) 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 (*φ*0 = 90°); the second peak corresponds to the incidence angle *φ*0 = cos-1(1/1.5) = 48.19°. The calculation of the current density distribution on the contour of each element is of practical interest, especially for the situations when the results obtained for infinite gratings are used for finite gratings. It is reasonable to assume that the near equal current distribution on all elements of the finite grating is a plausible argument to treat such grating as a fragment of the

In our calculations, we fix the number of the strips *N* =5 in the grating and calculate the current distribution on each element. The results are shown in Figure 10, where we used the param‐

Because of the symmetrical (for normal wave incidence) location of the elements in the array, the distribution | *J <sup>z</sup>*(*φ*)| will be identical for the strips, numerated by the indexes *n* =±1 and *n* =±2. 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 | *J <sup>z</sup>*(*φ*)|

<sup>0</sup> lim 2 , . *<sup>s</sup> RCS U*

pr

( ) <sup>2</sup>

= + (33)

*φ* =*φ*<sup>0</sup> + *π*, where *φ*0 is angle of incidence of the incident plane wave, via

r

infinite grating. This idea was used, for example, by Kalhor in [49].

®¥

**Figure 10.** | *J <sup>z</sup>*(*φ*)| at the strips *n* = −2, 1, 0 in the 5-element array (*k* =*π*, *t* / *w* =0.1, *d* =1.5).

 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 | *J <sup>z</sup>*(*φ*)|. 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 | *J <sup>z</sup>*(*φ*)| on the number of strips in the array is quite weak.

The scattered pattern in the direction *φ* is defined as

$$SP(\rho) = \lim\_{\rho \to \infty} \left( \left| \mathcal{U}^{\circ}(\rho, \rho) \right| / \max\_{\phi \in \{-x, x\}} \left| \mathcal{U}^{\circ}(\rho, \rho) \right| \right). \tag{34}$$

The distribution of the scattered field in the far-field zone is shown in Figures 11-13.

**Figure 11.** Scattering pattern of the 3-, 5- and 9-strip array *φ*<sup>0</sup> =90 , *k* =*π*, *t* / *w* =0.1, *d* =1.5.

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, *φ*<sup>0</sup> =30 , and wave number *k* =*kw* =2. With a small number of strips in the array, the shape of the main beam is not symmetric (the case *N* =3, 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 *N* =9.

the substantially improved shape of the main scattering beam when element number increases to N = 9 . Rigorous Approach to Analysis of Two-Dimensional Potential Problems, Wave Propagation and Scattering... http://dx.doi.org/10.5772/61287 197

beam is not symmetric (the case N = 3 , 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

**Figure 12.** Scattering pattern of the 3-, 5- and 9-strip array (*φ*<sup>0</sup> =30 , *k* =2*π*, *t* / *w* =0.1, *d* =1.5).

 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 | *J <sup>z</sup>*(*φ*)|. 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

( ) [ ,]

r j

= (34)

j pp

( ) lim ( , ) / max ( , ) . *s s SP U U*

The distribution of the scattered field in the far-field zone is shown in Figures 11-13.

 rj

**Figure 11.** Scattering pattern of the 3-, 5- and 9-strip array *φ*<sup>0</sup> =90 , *k* =*π*, *t* / *w* =0.1, *d* =1.5.

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, *φ*<sup>0</sup> =30 , and wave number *k* =*kw* =2. With a small number of strips in the array, the shape of the main beam is not symmetric (the case *N* =3, 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 *N* =9.

®¥ Î -

of | *J <sup>z</sup>*(*φ*)| on the number of strips in the array is quite weak.

r

The scattered pattern in the direction *φ* is defined as

196 Advanced Electromagnetic Waves

j

−30 **Figure 13.** Scattering pattern in dB of the 3-, 5- and 9-strip array (*φ*<sup>0</sup> =30 , *k* =2*π*, *t* / *w* =0.1, *d* =1.5).

0 50 100 150 200 250 300 350

�,degrees

Figure 13. Scattering pattern in dB of the 3-, 5- and 9-strip array ( <sup>0</sup> ϕ π == == 30 , 2 , / 0.1, 1.5 k tw d

−40

−35

−25

−20

Normalized SP,dB

−15

−10

−5

0

20

).

Next we consider the frequency dependence *RCS*(*k*) of the RCS on wavenumber. Setting the array parameters: *N* =3, 5, 9 *w* =1, *t* =0.1, *d* =1.5, *φ*<sup>0</sup> =90 , we calculate the function *RCS*(*k*) in the range 0≤*k* ≤25 (see Figure 14).

It is worth noting that extremely high values of the function *RCS*(*k*) are explained by its normalization (the single element width was chosen to be a characteristic parameter, so that *k* =*kw*. For the 9-element array analysed above the total width *W* of the array is *W* =9*w* + 8(*d* −*w*)=13 units (all sizes are related to the strip width); hence, *kW* =13*kw*. For comparison with the case of a single element case, the RCS should be scaled by the total width of array.

**Figure 14.** RCS versus relative wave number: *N* =3, 5, 9;*φ*<sup>0</sup> =90 , *t* / *w* =0.1, *d* =1.5.

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 *d* =0.4. At the lower values of the relative wave number (0≤*kw* ≤1.8), there is no influence of the nonsymmetrical location of the central strip on the value of the RCS. At the value *kw* =1.8, the shift *g* =0.9−0.5 (see Figure 15) becomes slightly greater than the wave size *λ* / 9. 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 *λ* / 9.

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 *w* greater than its height *h* (*h* / *w* <1); otherwise, it is more reasonable to call such a

**Figure 15.** RCS of the 9-element periodic array (black) and array with shifted central strip (red): *N* =9;*φ*<sup>0</sup> =90 , *t* / *w* =0.1, *d* =1.5.

structure a rectangular or square bar (*h* / *w* ≥1). The ratio *t* =*h* / *w* =1 represents a threshold value; the parameters *w* and *h* are better described as height and width as shown in Figure 16.

**Figure 16.** Schematic view of the strip thickening.

Next we consider the frequency dependence *RCS*(*k*) of the RCS on wavenumber. Setting the array parameters: *N* =3, 5, 9 *w* =1, *t* =0.1, *d* =1.5, *φ*<sup>0</sup> =90 , we calculate the function *RCS*(*k*) in

It is worth noting that extremely high values of the function *RCS*(*k*) are explained by its normalization (the single element width was chosen to be a characteristic parameter, so that *k* =*kw*. For the 9-element array analysed above the total width *W* of the array is *W* =9*w* + 8(*d* −*w*)=13 units (all sizes are related to the strip width); hence, *kW* =13*kw*. For comparison with the case of a single element case, the RCS should be scaled by the total width

**Figure 14.** RCS versus relative wave number: *N* =3, 5, 9;*φ*<sup>0</sup> =90 , *t* / *w* =0.1, *d* =1.5.

when the shift does not exceed *λ* / 9.

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 *d* =0.4. At the lower values of the relative wave number (0≤*kw* ≤1.8), there is no influence of the nonsymmetrical location of the central strip on the value of the RCS. At the value *kw* =1.8, the shift *g* =0.9−0.5 (see Figure 15) becomes slightly greater than the wave size *λ* / 9. Hence, we can conclude that the disturbance in the location of the strip in the array is insignificant on the RCS

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 *w* greater than its height *h* (*h* / *w* <1); otherwise, it is more reasonable to call such a

the range 0≤*k* ≤25 (see Figure 14).

198 Advanced Electromagnetic Waves

of array.

Here we consider normal wave incidence *φ*<sup>0</sup> =90 on the array of strips with the fixed param‐ eters: *w* =1, *d* / *w* =3, *k* ≡*kw* =*π*, 2*π* and 5*π*. Starting from the relative strip thickness *t* =*h* / *w* =0.05, we consider the dependence of RCS on *t* for a 3-element array in the range 0.05≤*t* =*h* / *w* ≤3. The results of these calculations are presented in Figure 17 (*k* =*π*), Figure 18 (*k* =2*π*) and Figure 19 (*k* =5*π*).

**Figure 17.** RCS of the 3-element array versus relative strip thickness *t* (*φ*<sup>0</sup> =90 , *k* =*π*, *w* =1, *d* =3).

**Figure 18.** RCS of the 3-element array versus relative strip thickness *t* (*φ*<sup>0</sup> =90 , *k* =2*π*, *w* =1, *d* =3).

**Figure 19.** RCS of the 3-element array versus relative strip thickness *t* (*φ*<sup>0</sup> =90 , *k* =5*π*, *w* =1, *d* =3).

Surprisingly the RCS is fairly insensitive to substantial thickening of the original horizontal thin strip (*t* =*h* / *w* =0.05), even at the extreme transformation of the strip into a vertical rectan‐ gular cylinder with *t* =*h* / *w* =3. The difference between the maximum value *RCS*(*t*) and its minimum value for the wavenumbers *k* =*π* and *k* =2*π* does not exceed 7.4*%*. In the case when *k* =5*π*, this difference increases to 8.3*%*.

### *4.2.2. Inclined arrays of thick strips*

**Figure 17.** RCS of the 3-element array versus relative strip thickness *t* (*φ*<sup>0</sup> =90 , *k* =*π*, *w* =1, *d* =3).

200 Advanced Electromagnetic Waves

**Figure 18.** RCS of the 3-element array versus relative strip thickness *t* (*φ*<sup>0</sup> =90 , *k* =2*π*, *w* =1, *d* =3).

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).

**Figure 20.** Truncated corner reflector (*φ*<sup>0</sup> =90 ): (a) isolated reflector and (b) array of three reflectors.

Here we only consider normal wave incidence, so we normalize the function *RCS*(*k*) by the characteristic maximum size of the structure. We define geometric parameters *W* for the single reflector and *W* ′ for the array of the corner reflectors (see Figure 20) by *W* =*d* + (*w* + *t*) / 2, *W* ′ =3*W* , where *d* is the distance between strip centres, *w* and *t* are the single strip relative width and thickness, respectively. We calculate the dependence *RCS*(*kW* ) for the three values *d* =0.8, 1.2, 1.6. As in previous calculations, we take *w* =1, *t* =0.1. The Epolarized plane wave excites the truncated corner reflector with incident angle *φ*<sup>0</sup> =90° . The frequency dependence of the RCS in the range 0<*kW* ≤20 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 *d* =0.8, 1.2, 1.6 ; 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 (*d* =0.8, 1.2) of the reflectors preserves the regular oscillatory behaviour of the *RCS*(*kW* ), in contrast to more separated spacing (*d* =1.6), where some peaks in the RCS graph (red, Figure 22) become suppressed.

**Figure 21.** *RCS*(*kW* ) for the single corner reflector: *φ*<sup>0</sup> =90 , *w* =1, *t* =0.1.

**Figure 22.** *RCS*(*kW* ′ ) for the array of three corner reflectors: *φ*<sup>0</sup> =90 , *w* =1, *t* =0.1.

26

differently profiled conductors and scattering problems for the arrays of thick strips illustrate

the efficiency and reliability of the method.

#### Figure 22. RCS kW( )′ for the array of three corner reflectors: <sup>0</sup> <sup>ϕ</sup> = == 90 , 1, 0.1 w t **5. Conclusion**

5. Conclusion

Here we only consider normal wave incidence, so we normalize the function *RCS*(*k*) by the characteristic maximum size of the structure. We define geometric parameters *W* for the single

single strip relative width and thickness, respectively. We calculate the dependence *RCS*(*kW* ) for the three values *d* =0.8, 1.2, 1.6. As in previous calculations, we take *w* =1, *t* =0.1. The Epolarized plane wave excites the truncated corner reflector with incident angle *φ*<sup>0</sup> =90° . The frequency dependence of the RCS in the range 0<*kW* ≤20 for the single truncated corner

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 *d* =0.8, 1.2, 1.6 ; 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 (*d* =0.8, 1.2) of the reflectors preserves the regular oscillatory behaviour of the *RCS*(*kW* ), in contrast to more separated spacing (*d* =1.6),

where some peaks in the RCS graph (red, Figure 22) become suppressed.

**Figure 21.** *RCS*(*kW* ) for the single corner reflector: *φ*<sup>0</sup> =90 , *w* =1, *t* =0.1.

for the array of the corner reflectors (see Figure 20) by

=3*W* , where *d* is the distance between strip centres, *w* and *t* are the

reflector and *W* ′

202 Advanced Electromagnetic Waves

*W* =*d* + (*w* + *t*) / 2, *W* ′

reflector is shown in Figure 21.

In this chapter, a rigorous approach to the numerical analysis of the multi-conductor 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.

.

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 Ntr . 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 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 *Ntr*. 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.
