The formulas of the distributed surface impedance
Abstract
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 antennafeeder devices, including combined vibratorslot structures. Widespread occurrence of such radiators is an objective prerequisite for theoretical analysis of their electrodynamic characteristics. During the 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, the hybrid analyticnumerical approaches, and the direct numerical techniques for electrodynamic analysis of such radiators. 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 antennafeeder 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 spatialenergy, and spatialpolarization 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 vibratorslot structures, should be created. This chapter presents the methodological basis for application of the generalized method of induced EMMF for the analysis of electrodynamic characteristics of the combined vibratorslot 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. 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 numericalanalytical 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 vibratorslot 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.
Keywords
 Waves excitation
 thin impedance vibrators
 narrow slots
 vibratorslot structures
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 antennafeeder devices, including combined vibratorslot structures [14]. 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 [57], the hybrid analyticnumerical approaches [810], 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 antennafeeder 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 spatialenergy and spatialpolarization 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 vibratorslot structures, should be created [1220].
Mathematical modeling of antennafeeder 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 antennafeeder 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 electromagnetomotive 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 vibratorslot 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 finitesize material bodies in two electrodynamic volumes coupled by holes cut in their common boundary. Suppose that there exists some arbitrary volume
To solve the abovementioned problem we express the electromagnetic fields in volumes
Here
Interpretation of the fields in the lefthand 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 wellconducting 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
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 (highfrequency 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 (lowfrequency or quasistatic 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 crosssection 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 slotvibrator systems can be generalized for multielement 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
Now we take into account that
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:
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 “quasionedimensional” 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 quasionedimensional analog of the exact integral equation with kernel (11) as starting point for the analysis. Note that impedance
where
Here
Here
is the vibrator selffield 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
where
This system is equivalent to the equation (18) and represents the standard equations system unsolvable with respect to derivatives. The righthand sides of the equations are proportional to small parameter
where
is selffield 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 (




1  The solid metallic cylinder of the 


2  The dielectrical metalized cylinder with covering, made of the metal of the 


3  The metaldielectrical cylinder ( 


4  The magnetodielectrical metalized cylinder with the inner conducting cylinder with the radius 


5  The metallic cylinder with covering, made of magnetodielectric of the 


6  The metallic monofilar helix of the 


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
where
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
where
Since the current distribution (30) is now known we can calculate electrodynamic characteristics of an impedance vibrator. Thus, an input impedance
where
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 semiinfinite 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
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
is selffield of the slot in infinite perfectly conducting plane,
is selffield 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
where
Assuming, as in Section 4,
where
is the slot total selffield, 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
where
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
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
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
For this configuration the system of integral equations relative to electrical currents at the vibrators
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
Here
where
The energy characteristics of the vibratorslot system: the reflection and transmission coefficients,
Let us consider several distribution functions for the surface impedance along the vibrator, namely: 1)
with the distribution function 2) as
and with the distribution function 3) as
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 singlemode 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 vibratorslot structures and coupled electrodynamic volumes expressions for
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 vibratorslot 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 vibratorslot 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 numericalanalytical 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 vibratorslot 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.
References
 1.
Elliott R.S. Antenna Theory and Design. Hoboken: John Wiley & Sons; 2003.  2.
Weiner M.M. Monopole Antennas. New York: Marcel Dekker; 2003.  3.
Nesterenko M.V., Katrich V.A., Penkin Yu.M, Berdnik S.L. Analytical and Hybrid Methods in Theory of SlotHole Coupling of Electrodynamic Volumes. New York: Springer Science + Business Media; 2008.  4.
Nesterenko M.V., Katrich V.A., Penkin Yu.M., Dakhov V.M., Berdnik S.L. Thin Impedance Vibrators. Theory and Applications. New York: Springer Science + Business Media; 2011.  5.
King R.W.P. The Theory of Linear Antennas. Cambr.Mass.: Harvard University Press; 1956.  6.
Nesterenko M.V., Katrich V.A. The asymptotic solution of an integral equation for magnetic current in a problem of waveguides coupling through narrow slots. Progress In Electromagnetics Research 2006;57 101129.  7.
Nesterenko M.V. Analytical methods in the theory of thin impedance vibrators. Progress In Electromagnetics Research B 2010;21 299328.  8.
Nesterenko M.V., Katrich V.A., Penkin Yu.M., Berdnik S.L. Analytical methods in theory of slothole coupling of electrodynamics volumes. Progress In Electromagnetics Research 2007;70 79174.  9.
Nesterenko M.V., Katrich V.A., Dakhov V.M., Berdnik S.L. Impedance vibrator with arbitrary point of excitation. Progress In Electromagnetics Research B 2008;5 275290.  10.
Nesterenko M.V., Katrich V.A., Berdnik S.L., Penkin Yu.M., Dakhov V.M. Application of the generalized method of induced EMF for investigation of characteristics of thin impedance vibrators. Progress In Electromagnetics Research B 2010;26 149178.  11.
Harrington R.F. Field Computation by Moment Methods, New York: IEEE Press ; 1993.  12.
Butler C.M., Umashankar K.R. Electromagnetic excitation of a wire through an apertureperforated conducting screen. IEEE Trans. Antennas and Propagat. 1976;24 456462.  13.
Itoh K., Matsumoto T. Theoretical analysis of mutual coupling between slot and unipole antennas. IEICE Trans. Commun. 1978;J61B 391397.  14.
Harrington R.F. Resonant behavior of a small aperture backed by a conducting body. IEEE Trans. Antennas and Propagat. 1982;30 205212.  15.
Naiheng Y., Harrington R.F. Electromagnetic coupling to an infinite wire through a slot in a conducting plane. IEEE Trans. Antennas and Propagat. 1983;31 310316.  16.
His S.W., Harrington R.F., Mautz J.R. Electromagnetic coupling to a conducting wire behind an aperture of arbitrary size and shape. IEEE Trans. Antennas and Propagat. 1985;33 581587.  17.
Park S., Hirokawa J., Ando M. Simple analysis of a slot and a reflectioncanceling post in a rectangular waveguide using only the axial uniform currents on the post surface. IEICE Trans. Commun. 2003;E86B 24822487.  18.
Kim K.C., Lim S.M., Kim M.S. Reduction of electromagnetic penetration through narrow slots in conducting screen by two parallel wires. IEICE Trans. Commun. 2005;E88B 17431745.  19.
Nesterenko M.V., Katrich V.A., Penkin Yu.M., Berdnik S.L., Kijko V.I. Combined vibratorslot structures in electrodynamic volumes. Progress In Electromagnetics Research B 2012;37 237256.  20.
Nesterenko M.V., Katrich V.A., Penkin D.Yu., Berdnik S.L., Kijko V.I. Electromagnetic waves scattering and radiation by vibratorslot structure in a rectangular waveguide. Progress In Electromagnetics Research M. 2012;24 6984.  21.
Lamensdorf D. An experimental investigation of dielectriccoated antennas. EEE Trans. Antennas and Propagat. 1967;15 767771.  22.
Bretones A.R., Martín R.G., García I.S. Timedomain analysis of magneticcoated wire antennas. IEEE Trans. Antennas and Propagat. 1995;43 591596.