## 1. Introduction

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 published 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 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 parameters, 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 this connection it should be noted that formation of analytical concepts of electrodynamic analysis extending the capabilities of physically correct mathematical models for new classes of boundary problems is always an important problem.

This chapter presents the methodological basis of a new approach to solving the electrodynamic problems associated with combined vibrator–slot structures, defined as a generalized method of induced electro-magneto-motive forces (EMMF). This approach is based on the classical method of induced EMMF, i.e, basis functions, approximating the currents along the vibrator and slot elements, are obtained in advance as analytical solutions of key problems, formulated as integral equations for the currents by the asymptotic averaging method. Bearing this in mind, we present here solutions of two key problems: a single impedance vibrator and slot scatterer in a waveguide, obtained by averaging method, and then solve a problem for the multielement vibrator-slot structures by generalized method of induced EMMF.

## 2. Problem formulation and initial integral equations

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

To solve the above-mentioned problem we express the electromagnetic fields in volumes

(1) |

Here

Interpretation of the fields in the left-hand side of equations (1) depends upon position of an observation point

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

where

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

(3) |

where

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.

## 3. Integral equations for electric and magnetic currents in thin impedance vibrators and narrow slots

A straightforward solution of the system (3) for the material objects with irregular surface shape and for holes with arbitrary geometry may often be impossible due to the known mathematical difficulties. However, the solution is sufficiently simplified for thin impedance vibrators and narrow slots, i.e. cylinders, which cross-section perimeter is small as compared to their length and the wavelength in the surrounding media and for holes, which one dimension satisfy the analogous conditions [19,20]. The approach used in [19,20] for the analysis of slot-vibrator systems can be generalized for multi-element systems. In addition, the boundary condition (2) can be extended for cylindrical vibrator surfaces with an arbitrary distribution of complex impedance regardless of the exciting field structure and electrophysical characteristics of vibrator material [4].

For thin vibrators made of circular cylindrical wire and narrow straight slots the equation system (3) can be easily simplified using inequalities

where

where

and the unknown currents

where upper indexes *e* and *m* are omitted.

Now we take into account that

(8) |

Here

For solitary vibrator or slot as well as for the absence of electromagnetic interaction between them, the system (8) splits into two independent equations:

(10) |

Here

Solution of the integral equation with the exact kernel expressions (11) and (12) may be very difficult, therefore we will use approximate expressions, the so called “quasi-one-dimensional” kernels [5,15]

derived with the assumption that source points belong to the geometric axes of the vibrator and slot while observation points belong to vibrator surface and to slot axis, having coordinates

Since the form of the Green’s functions was not specified, the equations (8) are valid for any electrodynamic volumes, provided that the Green’s functions for any electrodynamic volumes are known or can be constructed. Although the boundary between the volumes

## 4. Solution of integral equation for current in an impedance vibrator, located in unbounded free space

Let us use the equation (9) with the approximate kernel (13), being a quasi-one-dimensional analog of the exact integral equation with kernel (11) as starting point for the analysis. Note that impedance

where

Here

Here

(19) |

is the vibrator self-field in free space.

To find the approximate analytic solution of equation (18) we will use the asymptotic averaging method. The basic principles of the method are presented in [3,4]. To reduce the equation (18) to a standard equation system with a small parameter in compliance with the method of variation of constants we will change variables

(20) |

where

(21) |

This system is equivalent to the equation (18) and represents the standard equations system unsolvable with respect to derivatives. The right-hand sides of the equations are proportional to small parameter *partial averaging* means that averaging operator acts on all terms, but containing

(22) |

where

is self-field of the vibrator (19), averaged over its length.

We will seek the solution of the equations system (22) in the form

Then, substitution (24) into (22) gives

Then we find

where

For electrically thin vibrators (

The formulas have been obtained in the frames of impedance conception [4], and they are just for thin cylinders both of infinite and finite extension, located in free space. It is necessary to introduce the multiplier

The constants

(27) |

where

(28) |

It is evident that if an impedance vibrator is located in restricted volume

Let us consider a problem of vibrator excitation at its geometric center by a lumped EMF with amplitude

where

Here

(31) |

where

Since the current distribution (30) is now known we can calculate electrodynamic characteristics of an impedance vibrator. Thus, an input impedance

where

(33) |

Note, that an input admittance

To confirm the validity of the above analytical formulas we present the results of a comparative analysis of calculated and experimental data available in the literature. Figure 2 and Figure 3 show the graphs of the input admittance for two realizations of surface impedance: 1) metal wire (radius

## 5. Solution of equation for current in a slot between two semi-infinite rectangular waveguides

Now let us solve the second key problem. Let a resonant iris is placed in infinite hollow

A starting point for the analysis is equation (10) written as (index

(34) |

where

Isolating the logarithmic singularity in the kernel of equation (34) as in (17), we reduce the equation (34) to an integral equation with small parameter

Here

(36) |

is self-field of the slot in infinite perfectly conducting plane,

(37) |

is self-field of the slot, which takes into account multiple reflection from walls of volumes.

To solve the equation (35) by averaging method we change the variable according to (20) and obtain the standard system of integral equations relative to new unknown functions

(38) |

where

Assuming, as in Section 4,

where

is the slot total self-field, averaged over the slot length.

Solving the system (39), we obtain the general asymptotic expression for the current in narrow slot, located in arbitrary position relative to the walls of coupling volumes

To determine constants

(42) |

where

(43) |

which are completely defined by the Green’s functions of the coupling volumes.

Supposing that dominant wave

The symmetric and antisymmetric components of the slot current, relative to the slot center

(45) |

where

Reflection and transmission coefficients,

where

Figure 5 shows the theoretical and experimental wavelength dependences of power reflection coefficient

Note that a comparative analysis of the analytical solution of key problems is not limited only by the examples presented above. Thus, the solution for current in the impedance vibrator, located in free space, was preliminary compared with the known approximate analytical solutions of integral equations. The adequacy of the constructed mathematical models to real physical processes and the reliability of simulation results has been also confirmed by comparative calculations, obtained by the numerical method of moments and other methods, in particular, by the finite element method implemented in the software package *Ansoft HFSS*.

## 6. Combined vibrator–slot structures

Now let us consider a problem of electromagnetic waves excitation by a narrow straight transverse slot in the broad wall of rectangular waveguide with a two passive impedance vibrators in it.

Let a fundamental wave *Wg*”. Two thin nonsymmetrical vibrators (monopoles) with variable surface impedance are located in a waveguide with cross-section *Hs*”. The vibrators radiuses and lengths are

For this configuration the system of integral equations relative to electrical currents at the vibrators

(47) |

Here

We will seek the solution of equations system (47) by a generalized method of induced EMMF [19,20], using functions

In accordance with the generalized method of induced EMMF, we multiply equation (47a) by the function

(48) |

Here

(49) |

(51) |

(54) |

where

The energy characteristics of the vibrator-slot system: the reflection and transmission coefficients,

Let us consider several distribution functions for the surface impedance along the vibrator, namely: 1)

(58) |

with the distribution function 2) as

(59) |

and with the distribution function 3) as

(60) |

Since the formulas for

Figures 7, 8 shows the wavelength dependences of the radiation coefficient, modules of the reflection and transmission coefficients in the wavelength range of the waveguide single-mode regime, obtained using the following common parameters:

The choice of slot dimensions was stipulated by its natural resonance at the average wavelength of the waveguide frequency range

As might be expected from physical considerations, displacement of the impedance vibrator along the longitudinal axis of the waveguide at a distance

For the arbitrary vibrator-slot structures and coupled electrodynamic volumes expressions for

(61) |

where

Note once more that for arbitrary orientations of the vibrator, or the slot relative to the waveguide walls, or for another impressed field sources, the expressions (61) should be used to determine the distribution functions of electric and magnetic currents in the vibrator and slot. For example, for the longitudinal slot in the broad wall of waveguide, i.e. if axes

If vibrator is excited at its base by voltage

## 7. Conclusion

This chapter presents the methodological basis for application of the generalized method of induced EMMF for the analysis of electrodynamic characteristics of the combined vibrator-slot structures. Characteristic feature of the generalization to a new class of approximating functions consists in using them as a function of the current distributions along the impedance vibrator and slot elements; these distributions are derived as the asymptotic solution of integral equations for the current (key problems) by the method of averaging. Comparison of theoretical and experimental curves indicates that the solution of integral equations for combined vibrator-slot structures by the generalized method of induced EMMF with approximating functions for the currents in the impedance vibrator and the slot, obtained by averaging method is quite legitimate. It should be noted that for simple structures similar to that considered in the model problem, the proposed approach yields an analytic solution of the electrodynamic problem. For more complex structures, the method may be used to design effective numerical-analytical algorithms for their analyses.

The demonstrative simulation (the comparative analysis of all electrodynamic characteristics in the operating frequencies range) has confirmed the validity of the proposed 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 practically important problems, concerning the application of single impedance vibrator, including irregular vibrator, the systems of such vibrators and narrow slots. And this is a natural step in the further development of the general fundamental theory of linear radiators of electric and magnetic types.