**2. Problem formulation and initial integral equations**

The demonstrative simulation (the comparative analysis of all electrodynamic charac‐ teristics in the operating frequencies range) has confirmed the validity of the pro‐ posed generalized method of induced EMMF for analysis of vibrator-slot systems with rather arbitrary structure (within accepted assumptions). Here, as examples, some fragments of this comparative analysis were presented. This method retains all benefits of analytical methods as compared with direct numerical methods and allows to expand significantly the boundaries of numerical and analytical studies of practi‐ cally important problems, concerning the application of single impedance vibrator, in‐

**Keywords:** Waves excitation, thin impedance vibrators, narrow slots, vibrator-slot

At present, linear vibrator and slot radiators, i.e. radiators of electric and magnetic type, respectively, are widely used as separate receiver and transmitter structures, elements of antenna systems, and antenna-feeder devices, including combined vibrator-slot structures [1-4]. Widespread occurrence of such radiators is an objective prerequisite for theoretical analysis of their electrodynamic characteristics. During last decades researchers have pub‐ lished results which make it possible to create a modern theory of thin vibrator and narrow slot radiators. This theory combines the fundamental asymptotic methods for determining the single radiator characteristics [5-7], the hybrid analytic-numerical approaches [8-10], and the direct numerical techniques for electrodynamic analysis of such radiators [11]. However, the electrodynamics of single linear electric and magnetic radiators is far from been completed. It may be explained by further development of modern antenna techniques and antenna-feeder devices which can be characterized by such features as multielement structures, integration and modification of structural units to minimize their mass and dimensions and to ensure electromagnetic compatibility of radio aids, application of metamaterials, formation of required spatial-energy and spatial-polarization distributions of electromagnetic fields in various nondissipative and dissipative media. To solve these tasks electric and magnetic radiators, based on various impedance structures with irregular geometric or electrophysical

cluding irregular vibrator, the systems of such vibrators, and narrow slots.

parameters, and on combined vibrator-slot structures, should be created [12-20].

Mathematical modeling of antenna-feeder devices requires multiparametric optimization of electrodynamic problem solution and, hence, effective computational resources and software. Therefore, in spite of rapid growth of computer potential, there exists a necessity to develop new effective methods of electrodynamic analysis of antenna-feeder systems, being created with linear vibrator and slot structures with arbitrary geometric and electrophysical parame‐ ters, satisfying modern versatile requirements, and widening their application in various spheres. Efficiency of mathematical modeling is defined by rigor of corresponding boundary problem definition and solution, by performance of computational algorithm, requiring minimal possible RAM space, and directly depends upon analytical formulation of the models. That is, the weightier is the analytical component of the method the grater is its efficiency. In

structures

148 Advanced Electromagnetic Waves

**1. Introduction**

Let us formulate the problem of electromagnetic fields excitation (scattering, radiation) by finite-size material bodies in two electrodynamic volumes coupled by holes cut in their common boundary. Suppose that there exists some arbitrary volume *V*1, bounded by a perfectly conducting, impedance, or partially impedance surface *S*1, some parts of which may be infinitely distant. The volume *V*1 is coupled with another arbitrary volume *V*2 through holes Σ*<sup>n</sup>* (*n* =1, 2...*N* ), cut in the surface *S*1. The boundary between the volumes *V*1 and *V*2 in the regions around the coupling holes has an infinitely small thickness. Permittivity and perme‐ ability of the medium filling volumes *V*1 and *V*2 are *ε*1, *μ*1 and *ε*2, *μ*2, respectively. Material bodies, enclosed in local volumes *Vm*<sup>1</sup> (*m*<sup>1</sup> =1, 2, ...*M*1) and *Vm*<sup>2</sup> (*m*<sup>2</sup> =1, 2, ...*M*2), bounded by smooth closed surfaces *Sm*<sup>1</sup> and *Sm*<sup>2</sup> , are allocated in the volumes *V*1 and *V*2, respectively. The bodies have homogeneous material parameters: permittivity *εm*<sup>1</sup> , *εm*<sup>2</sup> , permeability *μm*<sup>1</sup> , *μm*<sup>2</sup> , and conductivity *σm*<sup>1</sup> , *σm*<sup>2</sup> . The fields of extraneous sources can be specified as the electromagnetic wave fields, incident on the bodies and the holes (scattering problem), or as fields of electro‐ motive forces, applied to the bodies (radiation problem), or as combination of these fields. Without loss of generality, we assume that electromagnetic fields of extraneous sources {*E* → 0(*r* <sup>→</sup> ), *H* → 0(*r* <sup>→</sup> )} exist only in the volume *V*1. The fields {*E* → 0(*r* <sup>→</sup> ), *H* → 0(*r* <sup>→</sup> )} depend on the time *t* as *eiω<sup>t</sup>* (*r* <sup>→</sup> is the radius vector of the observation point, *ω* =2*πf* is an circular frequency and *f* is frequency, measured in Hertz). We seek the electromagnetic fields {*E* → *V*1 (*r* <sup>→</sup> ), *H* → *V*1 (*r* <sup>→</sup> )} and {*E* → *V*2 (*r* <sup>→</sup> ), *H* → *V*2 (*r* <sup>→</sup> )} in the volumes *V*1 and *V*2, satisfying Maxwell's equations and boundary conditions on the surfaces *Sm*<sup>1</sup> , *Sm*<sup>2</sup> , Σ*n*, *S*1 and *S*2 (Figure 1).

To solve the above-mentioned problem we express the electromagnetic fields in volumes *V*<sup>1</sup> and *V*2 in terms of the tangential fields components on the surfaces *Sm*<sup>1</sup> , *Sm*<sup>2</sup> and Σ*n*. In the Gaussian CGS system of units, the electromagnetic fields can be represented by the wellknown Kirchhoff-Kotler integral equations [3,4]:

**Figure 1.** The problem geometry and notations

graddiv d rot d d 1 1 1 1 1 11 1 <sup>1</sup> <sup>1</sup> 1 1 1 1 11 1 1 1 <sup>1</sup> <sup>1</sup> 2 0 1 1 1 1 1 <sup>1</sup> <sup>ˆ</sup> ( ) ( ) ( ) ( , )[ , ( )] <sup>4</sup> <sup>1</sup> ˆ ˆ ( , )[ , ( )] (, ) , ( ) <sup>4</sup> *m m n M e V V m m Vm m m S <sup>M</sup> <sup>N</sup> m m V m m Vm m V n nVn n m n S E r Er k G rr n H r r ik G rr n E r r G rr n E r r* p e p = = = S =+ + ¢ ¢¢ ì ü ï ï - + ¢ ¢ ¢ ¢ ¢¢ é ù í ý ë û ï ï î þ å ò å å ò ò r r r r r rr r r r r r rr r r r rr r r r d graddiv d rot d 1 1 1 1 11 1 <sup>1</sup> <sup>1</sup> 1 1 1 1 1 1 11 1 <sup>1</sup> <sup>2</sup> 0 1 1 1 , ˆ (, ) , ( ) <sup>1</sup> () () ( ) <sup>4</sup> <sup>ˆ</sup> (, ) , ( ) <sup>1</sup> <sup>ˆ</sup> (, ) , ( ) <sup>4</sup> *m n m M m V m m Vm m m S <sup>V</sup> <sup>N</sup> <sup>m</sup> V n nVn n n e V m m Vm m S G rr n E r r H r Hr k ik G rr n E r r G rr n H r r* p m p = = S ì ü ¢ ¢¢ é ù ï ï ë û =+ + í ý ï ï <sup>+</sup> ¢ ¢¢ é ù ë û î þ <sup>+</sup> ¢ ¢¢ é ù ë û å ò å ò r rr r r r r r r r r rr r r r r rr r r r <sup>1</sup> <sup>1</sup> <sup>1</sup> 1 , *M m* = å ò (1)

graddiv d rot d d 2 2 2 2 2 22 2 <sup>2</sup> <sup>2</sup> 2 2 2 2 22 2 2 2 <sup>2</sup> <sup>2</sup> 2 2 2 2 1 1 1 <sup>1</sup> <sup>ˆ</sup> () ( ) (, ) , ( ) <sup>4</sup> <sup>1</sup> ˆ ˆ ( , )[ , ( )] (, ) , ( ) , <sup>4</sup> ( *m m n M e V V m m Vm m m S <sup>M</sup> <sup>N</sup> m m V m m Vm m V n nVn n m n S V E r k G rr n H r r ik G rr n E r r G rr n E r r H* p e p = = = S = + ¢ ¢¢ é ù ë û ì ü ï ï - + ¢ ¢ ¢ ¢ ¢¢ é ù í ý ë û ï ï î þ å ò å å ò ò r r r rr r r r r r rr r r r rr r r r <sup>r</sup> <sup>d</sup> graddiv d rot 2 2 2 2 22 2 <sup>2</sup> <sup>2</sup> 2 2 2 2 2 2 22 2 <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>2</sup> 2 2 1 1 ˆ (, ) , ( ) <sup>1</sup> )( ) <sup>4</sup> <sup>ˆ</sup> (, ) , ( ) <sup>1</sup> <sup>ˆ</sup> (, ) , ( ) . <sup>4</sup> *m n m M m V m m Vm m m S <sup>N</sup> <sup>m</sup> V n nVn n n M e V m m Vm m m S G rr n E r r r k ik G rr n E r r G rr n H r r* p m p = = S = ì ü ¢ ¢¢ é ù ï ï ë û = + í ý ï ï <sup>+</sup> ¢ ¢¢ é ù ë û î þ <sup>+</sup> ¢ ¢¢ é ù ë û å ò å ò å ò r rr r r r r r rr r r r r rr r r r

Here *k* =2*π* / *λ* is the wave number, *λ* is the free space wavelength, *k*<sup>1</sup> =*k ε*1*μ*<sup>1</sup> and *k*<sup>2</sup> =*k ε*2*μ*<sup>2</sup> are wave numbers in the media filling the volumes *V*1 and *V*2, respectively; *r* → ′ *<sup>m</sup>*1,*m*2,*<sup>n</sup>* are radius-vectors of sources allocated at the surfaces *Sm*<sup>1</sup> , *Sm*<sup>2</sup> and Σ*n* ; *n* → *<sup>m</sup>*1,*m*2,*<sup>n</sup>* are unit vectors of external normals to the surfaces; *G* ^ *V*1,*V*<sup>2</sup> *<sup>e</sup>* (*r* <sup>→</sup> , *r* → ′ ) and *G* ^ *V*1,*V*<sup>2</sup> *<sup>m</sup>* (*r* <sup>→</sup> , *r* → ′ ) are the electric and magnetic tensor Green's functions for Hertz's vector potentials in the coupled volumes satisfying the vector Helmholtz equation and the boundary conditions on surfaces *S*1 and *S*2. For the infinitely distant parts of surfaces *S*1 or *S*2 the boundary conditions for the Green's functions are transformed to the Sommerfeld's radiation condition.

Interpretation of the fields in the left-hand side of equations (1) depends upon position of an observation point *r* <sup>→</sup> . If the observation point *r* <sup>→</sup> belongs to the surfaces *Sm*<sup>1</sup> , *Sm*<sup>2</sup> or to the apertures Σ*n*, the fields *E* → (*r* <sup>→</sup> ) and *H* →(*r* <sup>→</sup> ) represent the same fields as in the integrals in the right-hand sides of equations (1). In this case, equations (1) are non-homogeneous linear integral Fredholm equations of the second kind, which are known to have the unique solution. If the observation point lies outside areas *Vm*<sup>1</sup> , *Vm*<sup>2</sup> and Σ*n*, the equations (1) become the equalities determining the total electromagnetic field by the field of specified extraneous sources. These equalities solve, in general terms, the problem of electromagnetic fields excitation by finite size obstacles if fields on the objects' surfaces are known. Certainly, to find these fields, the Fredholm integral equations should be solved beforehand.

graddiv d

*V m m Vm m V n nVn n*

r r rr r r r rr r r r

*m*

d

,

(1)

d

1 1 1 11 1

r rr r r r

*V m m Vm m*

r rr r r r

ˆ (, ) , ( )

ì ü ¢ ¢¢ é ù ï ï ë û

*G rr n E r r*

*V n nVn n*

*G rr n E r r*

ï ï <sup>+</sup> ¢ ¢¢ é ù ë û î þ

ë û

2 2 2 22 2

r rr r r r

*V m m Vm m*

ˆ (, ) , ( )

ì ü ¢ ¢¢ é ù ï ï ë û

*G rr n E r r*

r rr r r r

*V n nVn n*

*G rr n E r r*

ï ï <sup>+</sup> ¢ ¢¢ é ù ë û î þ

d

1 1

*G rr n E r r G rr n E r r*

*e*

rot d d

ï ï - + ¢ ¢ ¢ ¢ ¢¢ é ù í ý ë û ï ï î þ

1 1 1 11 1 1 1

1

å ò

*m S*

=+ + í ý

=

*M*

<sup>1</sup> <sup>2</sup>

<sup>1</sup> <sup>1</sup>

*m*

1

å ò

= S

,

graddiv d

*e*

rot d d

ï ï - + ¢ ¢ ¢ ¢ ¢¢ é ù í ý ë û ï ï î þ

<sup>1</sup> ˆ ˆ ( , )[ , ( )] (, ) , ( ) , <sup>4</sup>

*m*

2 2 2 22 2 2 2

ì ü

*V m m Vm m V n nVn n*

r r rr r r r rr r r r

2 2

*G rr n E r r G rr n E r r*

*n*

*n*

*m*

ì ü

1

**Figure 1.** The problem geometry and notations

p

150 Advanced Electromagnetic Waves

1

r r r r

p

2

*V*

*H*

(

r

rot

p

2

p m

*M*

<sup>2</sup> <sup>2</sup>

*m*

1

å ò

*m S*

=

p

0 1

rot d

<sup>1</sup> <sup>ˆ</sup> (, ) , ( ) <sup>4</sup>

+ ¢ ¢¢ é ù

*e*

*H r Hr k ik*

p m

0 1 1

*<sup>V</sup> <sup>N</sup> <sup>m</sup>*

r rr r r r <sup>1</sup>

*ik*

p e

<sup>1</sup> <sup>1</sup>

*m*

2

*ik*

p e

<sup>2</sup> <sup>2</sup>

2

*r k ik*

*e*

*S*

<sup>1</sup> <sup>1</sup> 1

*M*

*m* = å ò

1 1 1 1 1 11 1 <sup>1</sup> <sup>1</sup>

å ò

=

*M*

=+ + ¢ ¢¢

2

*V V m m Vm m m S*

<sup>1</sup> ˆ ˆ ( , )[ , ( )] (, ) , ( ) <sup>4</sup>

<sup>1</sup> () () ( ) <sup>4</sup> <sup>ˆ</sup> (, ) , ( )

2 2 2 2 2 22 2 <sup>2</sup> <sup>2</sup>

r r r rr r r r

*M*

<sup>1</sup> <sup>ˆ</sup> () ( ) (, ) , ( ) <sup>4</sup>

*E r k G rr n H r r*

=

*V V m m Vm m m S <sup>M</sup> <sup>N</sup> m m*

å ò

*m*

= + ¢ ¢¢ é ù

1 1

2 2 2 22 2

r rr r r r

*V m m Vm m*

*G rr n H r r*

å å ò ò

= = S

*m n S*

graddiv

<sup>1</sup> <sup>ˆ</sup> (, ) , ( ) . <sup>4</sup>

+ ¢ ¢¢ é ù

*m n*

<sup>r</sup> <sup>d</sup>

<sup>1</sup> <sup>2</sup> 2

<sup>1</sup> )( ) <sup>4</sup> <sup>ˆ</sup> (, ) , ( )

ë û

2

*m S*

=

å ò

*M*

<sup>2</sup> <sup>2</sup>

*m*

1

å ò

= S

*n*

*n*

*<sup>N</sup> <sup>m</sup>*

= + í ý

<sup>1</sup> <sup>ˆ</sup> ( ) ( ) ( ) ( , )[ , ( )] <sup>4</sup>

*E r Er k G rr n H r r*

r r r r r rr r r r

1 1

1 1

graddiv

1 1 1 11 1

ë û

2 2 2 1

*V m m Vm m*

*G rr n H r r*

å å ò ò

= = S

*m n S*

*m n*

*<sup>M</sup> <sup>N</sup> m m*

The equations (1) can be also used to solve electrodynamics problems if the fields on the material body surfaces can be defined by additional physical considerations. For example, if induced currents on well-conducting bodies (*σ* →*∞*) are concentrated near the body surface the skin layer thickness can be neglected and the well-known Leontovich-Shchukin approxi‐ mate impedance boundary condition becomes applicable [4]

$$\left[\vec{n}, \vec{E}(\vec{r})\right] = \overline{Z}\_{\mathbb{S}}(\vec{r}) \Big[\vec{n}, \Big[\vec{n}, \vec{H}(\vec{r})\Big]\Big],\tag{2}$$

where *Z*¯ *S* (*r* <sup>→</sup> )=*<sup>R</sup>*¯ *S* (*r* <sup>→</sup> ) <sup>+</sup> *iX*¯ *S* (*r* <sup>→</sup> )=*ZS* (*r* <sup>→</sup> )/ *Z*<sup>0</sup> is the distributed complex surface impedance, normal‐ ized to the characteristic free space impedance *Z*<sup>0</sup> =120*π* Ohm; the value of *Z*¯ *S* (*r* <sup>→</sup> ) may vary over the body surface. It is generally accepted that the boundary condition (2) are physically adequate under condition |*Z*¯ *S* (*r* <sup>→</sup> )| <sup>≪</sup>1. If |*Z*¯ *S* (*r* <sup>→</sup> )| →0, the boundary condition become that for the perfect conductor. In contrast to the limiting case of the perfect conductor, the impe‐ dance boundary condition allow to take into account losses in the real material. Since the relative error of (2) is of order |*Z*¯ *S* (*r* <sup>→</sup> )| 3, the inequality 0<sup>≤</sup> <sup>|</sup>*<sup>Z</sup>*¯ *S* (*r* <sup>→</sup> )| ≤0.4 must hold to obtain valid results by the mathematical model.

Using the impedance boundary condition (2) we can introduce a new unknown, density of surface currents. Let us perform such change of unknown in the equations (1). Without loss of generality, we carry the system of equations (1) the transition to the case when all the material bodies are located in volume *V*1. By placing the observation point on the surface *Sm* (index 1

is omitted) and using the continuity condition for the tangential components of the magnetic field on the holes Σ*n*, we obtain the system of integral equations relative to the density of surface currents: electric *J* → *m <sup>e</sup>* (*r* → *<sup>m</sup>*) at *Sm* and equivalent magnetic *J* → *n <sup>m</sup>*(*r* → *<sup>n</sup>*) at Σ*n*. The system can be presented as

rot d graddiv d rot d 1 1 1 11 1 2 0 1 1 1 1 0 ˆ () () (,) () <sup>1</sup> <sup>ˆ</sup> () ( ) (, ) ( ) <sup>1</sup> <sup>ˆ</sup> ( , ) ( )[ , ( )] , (a) <sup>4</sup> ( *n m <sup>N</sup> <sup>e</sup> m m Sq q q q V qn n n n n <sup>M</sup> e e q V qm mm m m S <sup>M</sup> m e V q m Sm m m m m m m S <sup>k</sup> Z rJr G rrJ r r E r k G rr J r r <sup>i</sup> G rr Z r n J r r H r* w we p = S = = + = ¢ ¢¢ =+ + ¢ ¢¢ + + ¢ ¢ ¢¢ å ò å ò å ò r r r r rr r r r r r rr r r r rr r r r r <sup>r</sup> <sup>r</sup> graddiv <sup>d</sup> graddiv d graddiv d rot 1 2 1 1 2 1 1 1 2 2 2 1 2 1 1 1 <sup>1</sup> <sup>ˆ</sup> ) ( ) (,) () <sup>1</sup> <sup>ˆ</sup> ( ) (,) () <sup>1</sup> <sup>ˆ</sup> ( ) ( , ) ( )[ , ( )] ˆ *n n m <sup>N</sup> m m p V pn n n n n <sup>N</sup> m m V pn n n n n <sup>M</sup> m e V p m Sm m m m m m m S e V k G rrJ r r <sup>i</sup> k G rrJ r r <sup>i</sup> k G rrZ r n J r r <sup>i</sup> <sup>k</sup> <sup>G</sup>* wm wm we w = S = S = + + ¢ ¢¢ + + ¢ ¢¢ = + - ¢ ¢ ¢¢ å ò å ò å ò r rr r r r rr r r r rr r r r r d 1 (, ) () , (b) *m <sup>M</sup> <sup>e</sup> pm mm m m S rr J r r* = <sup>å</sup> ¢ ¢¢ <sup>ò</sup> <sup>r</sup> rr r r (3)

where *q* =1, 2, ..., *m*, ..., *M* , *p* =1, 2, ..., *n*, ..., *N* , *J* → *m <sup>e</sup>* (*r* → *<sup>m</sup>*)= *<sup>c</sup>* <sup>4</sup>*<sup>π</sup> <sup>n</sup>* → *<sup>m</sup>*, *H* →(*r* → *<sup>m</sup>*) , *J* → *n <sup>m</sup>*(*r* → *<sup>n</sup>*)= *<sup>c</sup>* <sup>4</sup>*<sup>π</sup> <sup>n</sup>* → *<sup>n</sup>*, *E* → (*r* → *<sup>n</sup>*) , *c* is velocity of light in free space.

Thus, the problem of electromagnetic waves excitation by the impedance bodies of finite dimensions and by the coupling holes between two electrodynamic volumes is formulated as a rigorous boundary value problem of macroscopic electrodynamics, reduced to the system of integral equations for surface currents. Solution of this system is an independent problem, significant in its own right, since it often present considerable mathematical difficulties. If characteristic dimensions of an object are much greater than wavelength (high-frequency region) a solution is usually searched as series expansion in ascending power of inverse wave number. If dimensions of an object are less than wavelength (low-frequency or quasi-static region), representation of the unknown functions as series expansion in wave number powers reduces the problem to a sequence of electrostatic problems. Contrary to asymptotic cases, resonant region, where at least one dimension of an object is comparable with wavelength, is the most complex for analysis, and requires rigorous solution of field equations. It should be noted that, from the practical point of view, the resonant region is of exceptional interest for thin impedance vibrators and narrow slots.
