Open access peer-reviewed chapter

Electromagnetic Waves Excitation by Thin Impedance Vibrators and Narrow Slots in Electrodynamic Volumes

By Mikhail V. Nesterenko, Sergey L. Berdnik, Victor A. Katrich and Yuriy M. Penkin

Submitted: October 14th 2014Reviewed: July 8th 2015Published: November 18th 2015

DOI: 10.5772/61188

Downloaded: 742

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 antenna-feeder devices, including combined vibrator-slot 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 analytic-numerical 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 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. 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. 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.

Keywords

  • Waves excitation
  • thin impedance vibrators
  • narrow slots
  • vibrator-slot 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 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 V1, bounded by a perfectly conducting, impedance, or partially impedance surface S1, some parts of which may be infinitely distant. The volume V1is coupled with another arbitrary volume V2through holes Σn(n=1,2...N), cut in the surface S1. The boundary between the volumes V1and V2in the regions around the coupling holes has an infinitely small thickness. Permittivity and permeability of the medium filling volumes V1and V2are ε1,μ1and ε2,μ2, respectively. Material bodies, enclosed in local volumes Vm1(m1=1,2,...M1)and Vm2(m2=1,2,...M2), bounded by smooth closed surfaces Sm1and Sm2, are allocated in the volumes V1and V2, respectively. The bodies have homogeneous material parameters: permittivity εm1, εm2, permeability μm1, μm2, and conductivity σm1, σm2. 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 electromotive 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 {E0(r),H0(r)}exist only in the volume V1. The fields {E0(r),H0(r)}depend on the time tas eiωt(ris the radius vector of the observation point, ω=2πfis an circular frequency and fis frequency, measured in Hertz). We seek the electromagnetic fields {EV1(r),HV1(r)}and {EV2(r),HV2(r)}in the volumes V1and V2, satisfying Maxwell’s equations and boundary conditions on the surfaces Sm1, Sm2, Σn, S1and S2(Figure 1).

Figure 1.

The problem geometry and notations

To solve the above-mentioned problem we express the electromagnetic fields in volumes V1and V2in terms of the tangential fields components on the surfaces Sm1, Sm2and Σn. In the Gaussian CGS system of units, the electromagnetic fields can be represented by the well-known Kirchhoff-Kotler integral equations [3,4]:

EV1(r)=E0(r)+14πikε1(graddiv+k12)m1=1M1Sm1G^V1e(r,rm1)[nm1,HV1(rm1)]drm114πrot{m1=1M1Sm1G^V1m(r,rm1)[nm1,EV1(rm1)]drm1+n=1NΣnG^V1m(r,rn)[nn,EV1(rn)]drn},HV1(r)=H0(r)+14πikμ1(graddiv+k12){m1=1M1Sm1G^V1m(r,rm1)[nm1,EV1(rm1)]drm1+n=1NΣnG^V1m(r,rn)[nn,EV1(rn)]drn}+14πrotm1=1M1Sm1G^V1e(r,rm1)[nm1,HV1(rm1)]drm1,E1
EV2(r)=14πikε2(graddiv+k22)m2=1M2Sm2G^V2e(r,rm2)[nm2,HV2(rm2)]drm214πrot{m2=1M2Sm2G^V2m(r,rm2)[nm2,EV2(rm2)]drm2+n=1NΣnG^V2m(r,rn)[nn,EV2(rn)]drn},HV2(r)=14πikμ2(graddiv+k22){m2=1M2Sm2G^V2m(r,rm2)[nm2,EV2(rm2)]drm2+n=1NΣnG^V2m(r,rn)[nn,EV2(rn)]drn}+14πrotm2=1M2Sm2G^V2e(r,rm2)[nm2,HV2(rm2)]rm2.BB1

Here k=2π/λis the wave number, λis the free space wavelength, k1=kε1μ1and k2=kε2μ2are wave numbers in the media filling the volumes V1and V2, respectively; rm1,m2,nare radius-vectors of sources allocated at the surfaces Sm1, Sm2and Σn; nm1,m2,nare unit vectors of external normals to the surfaces; G^V1,V2e(r,r)and G^V1,V2m(r,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 S1and S2. For the infinitely distant parts of surfaces S1or S2the 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. If the observation point rbelongs to the surfaces Sm1, Sm2or to the apertures Σn, the fields E(r)and H(r)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 Vm1, Vm2and Σn, 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.

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 approximate impedance boundary condition becomes applicable [4]

[n,E(r)]=Z¯S(r)[n,[n,H(r)]],E2

where Z¯S(r)=R¯S(r)+iX¯S(r)=ZS(r)/Z0is the distributed complex surface impedance, normalized to the characteristic free space impedance Z0=120πOhm; the value of Z¯S(r)may vary over the body surface. It is generally accepted that the boundary condition (2) are physically adequate under condition |Z¯S(r)|1. If |Z¯S(r)|0, the boundary condition become that for the perfect conductor. In contrast to the limiting case of the perfect conductor, the impedance boundary condition allow to take into account losses in the real material. Since the relative error of (2) is of order |Z¯S(r)|3, the inequality 0|Z¯S(r)|0.4must 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 V1. 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 Σn, we obtain the system of integral equations relative to the density of surface currents: electric Jme(rm)at Smand equivalent magnetic Jnm(rn)at Σn. The system can be presented as

ZSq(rq)Jqe(rq)+kωrotn=1NΣnG^V1m(rq,rn)Jnm(rn)drn==E0(rq)+1iωε1(graddiv+k12)m=1MSmG^V1e(rq,rm)Jme(rm)drm++14πrotm=1MS11G^V1m(rq,rm)ZSm(rm)[nm,Jme(rm)]drm,(a)H0(rp)+1iωμ1(graddiv+k12)n=1NΣnG^V1m(rp,rn)Jnm(rn)drn+1iωμ2(graddiv+k22)n=1NΣnG^V2m(rp,rn)Jnm(rn)drn=1iωε1(graddiv+k12)m=1MSmG^V1m(rp,rm)ZSm(rm)[nm,Jme(rm)]drmkωrotm=1MSmG^V1e(rp,rm)Jme(rm)drm,(b)E3

where q=1,2,...,m,...,M, p=1,2,...,n,...,N, Jme(rm)=c4π[nm,H(rm)], Jnm(rn)=c4π[nn,E(rn)], cis 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.

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

rmLm<<1,rmλ1,2<<1,dn2Ln<<1,dnλ1,2<<1,E4

where rmis vibrator radius, Lmis vibrator length, dnis slot width, 2Lnis slot length, and λ1,2is wavelength in the corresponding media. The electric current induced on the vibrator surfaces and equivalent magnetic currents in the slots can be presented using the inequalities (4) as

Jme(rm)=esmJm(sm)ψm(ρm,φm),Jnm(rn)=esnJn(sn)χn(ξn),E5

where esmand esnare unit vectors directed along the vibrator and slot axis, respectively; smand snare local coordinates related to the vibrator and slot axes; ψm(ρm,φm)are functions of transverse (m) polar coordinates ρm, φmfor the vibrators; χn(ξn)are functions of transverse coordinates ξnfor the slots. The functions ψm(ρm,φm)and χn(ξn)satisfy the normality conditions

mψm(ρm,φm)ρmdρmdφm=1,ξnχn(ξn)dξn=1,E6

and the unknown currents Jm(sm)and Jn(sn)must satisfy the boundary conditions

Jm(±Lm)=0,Jn(±Ln)=0,E7

where upper indexes e and m are omitted.

Now we take into account that [nm,Jm(rm)]1according to inequalities (4) and project the equations (3a) and (3b) on the axes of the vibrators and slots, respectively, and arrive at a system of linear integral equations relative to the currents in the vibrators and slots

(d2dsq2+k12)m=1MLmLmJm(sm)GsmV1(sq,sm)dsmikesqrotn=1NLnLnJn(sn)GsnV1(sq,sn)dsn=iωε1[E0sq(sq)ziq(sq)Jq(sq)],1μ1(d2dsp2+k12)n=1NLnLnJn(sn)GsnV1(sp,sn)dsn+1μ2(d2dsp2+k22)n=1NLnLnJn(sn)GsnV2(sp,sn)dsn+ikesprotm=1MLmLmJm(sm)GsmV1(sp,sm)dsm=iωH0sp(sp).E8

Here zim(sm)are internal lineal impedance of the vibrators (ZSm(rm)=2πrmzim(rm)) measured in Ohm/m, E0sm(sm)and H0sn(sn)are projections of extraneous sources on the vibrators and slots axes, GsmV1(sm,n,sm)and GsnV1(2)(sm,n,sn)are components of the tensor Green’s functions in the volumes V1and V2.

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

(d2dsv2+k12)LvLvJv(sv)GsvV(sv,sv)dsv=iωε1E0sv(sv)+iωε1zi(sv)Jv(sv),E9
1μ1(d2dssl2+k12)LslLslJsl(ssl)GsslV1(ssl,ssl)dssl+1μ2(d2dssl2+k22)LslLslJsl(ssl)GsslV2(ssl,ssl)dssl=iωH0ssl(ssl).E10

Here esvand esslare unit vectors of vibrator and slot axes at the sources, and

GsvV(sv,sv)=ππeik1(svsv)2+[2rsin(φ/2)]2(svsv)2+[2rsin(φ/2)]2ψ(r,φ)rdφ,E11
GsslV1,2(ssl,ssl)=d/2d/2eik1,2(sslssl)2+(ξ)2(sslssl)2+(ξ)2χ(ξ)dξ.E12

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]

GsvV(sv,sv)=eik1(svsv)2+r2(svsv)2+r2,E13
GsslV1,2(ssl,ssl)=eik1,2(sslssl)2+(d/4)2(sslssl)2+(d/4)2E14

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 {ssl,ξ/2}. In that case the functions GsvV(sv,sv)and GsslV1,2(ssl,ssl)are everywhere continuous and equations for the currents are simplified significantly.

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 V1and V2initially was supposed to be of infinitesimal thickness, its actual thickness can be accounted for by introducing into the equations (8) an effective slot width, defined by the formula given in the Section 5.

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 zi(s)const, ε1=μ1=1, and index vis omitted. Thus, the equation may be written as

(d2ds2+k2)LLJ(s)eikR(s,s)R(s,s)ds=iωE0s(s)+iωziJ(s),E15

where R(s,s)=(ss)2+r2. Let us isolate the logarithmic singularity in the kernel of equation (15) by identical transformation

LLJ(s)eikR(s,s)R(s,s)ds=Ω(s)J(s)+LLJ(s)eikR(s,s)J(s)R(s,s)ds.E16

Here

Ω(s)=LLds(ss)2+r2=Ω+γ(s),E17

γ(s)=ln[(L+s)+(L+s)2+r2][(Ls)+(Ls)2+r2]4L2is a function, equal to zero at the vibrator center and reaching maximal value at its ends where the current in accordance with boundary condition (7) is equal to zero, Ω=2ln2Lris a large parameter. Then, equation (15) in view of (16) is transformed to integral equation with a small parameter

d2J(s)ds2+k2J(s)=α{iωE0s(s)+F[s,J(s)]iωziJ(s)}.E18

Here α=1Ω=12ln[r/(2L)]is a natural small parameter of the problem (|α|<<1),

F[s,J(s)]=dJ(s)dseikR(s,s)R(s,s)|LL+[d2J(s)ds2+k2J(s)]γ(s)+LL[d2J(s)ds2+k2J(s)]eikR(s,s)[d2J(s)ds2+k2J(s)]R(s,s)dsE19

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

J(s)=A(s)cosks+B(s)sinks,dJ(s)ds=A(s)ksinks+B(s)kcosks,(dA(s)dscosks+dB(s)dssinks=0),d2J(s)ds2+k2J(s)=dA(s)dssinks+dB(s)dscosks,E20

where A(s)and B(s)are new unknown functions. Then the equation (18) reduces to a system of integral equations

dA(s)ds=αk{iωE0s(s)+F[s,A(s),dA(s)ds,B(s),dB(s)ds]iωzi[A(s)cosks+B(s)sinks]}sinks,dB(s)ds=+αk{iωE0s(s)+F[s,A(s),dA(s)ds,B(s),dB(s)ds]iωzi[A(s)cosks+B(s)sinks]}cosks.E21

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 α, therefore, the functions A(s)and B(s)in the left-hand sides of the equations system (21) are slowly varying functions and the system can be solved by the asymptotic averaging method. Then, we replace the system (21) by the simplified system wherein assume dA(s)ds=0and dB(s)ds=0in rigth-hand members and carry out partial averaging over the explicit variable sto obtain the equations of first approximation. The term partial averaging means that averaging operator acts on all terms, but containing E0s(s)and it may be done for the system (21). The averaged system can be written as

dA¯(s)ds=α{iωkE0s(s)+F¯[s,A¯(s),B¯(s)]}sinks+χB¯(s),dB¯(s)ds=+α{iωkE0s(s)+F¯[s,A¯(s),B¯(s)]}cosksχA¯(s),E22

where χ=αiω2kzi,

F¯[s,A¯(s),B¯(s)]=[A¯(s)sinksB¯(s)cosks]eikR(s,s)R(s,s)|LLE23

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

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

A¯(s)=C1(s)cosχs+C2(s)sinχs,B¯(s)=C1(s)sinχs+C2(s)cosχs.E24

Then, substitution (24) into (22) gives

dC1(s)ds=α{iωkE0s(s)+F¯[s,C1,C2]}sin(k+χ)s,dC2(s)ds=+α{iωkE0s(s)+F¯[s,C1,C2]}cos(k+χ)s.E25

Then we find C1(s)and C2(s)by solving system (25), determine A¯(s)и B¯(s)from (24), and substitute them as approximating functions for the current into (20). Thus, the general asymptotic expression in parameter αfor the current in a thin impedance vibrator under arbitrary excitation may be presented as

J(s)=A¯(L)cos(k˜s+χL)+B¯(L)sin(k˜s+χL)+αLs{iωkE0s(s)+F¯[s,A¯,B¯]}sink˜(ss)ds,E26

where k˜=k+χ=k+i(α/r)Z¯S, Z¯S=R¯S+iX¯Sis the normalized complex surface impedance: Z¯S=2πrzi/Z0.

For electrically thin vibrators (|(kεμr)2ln(kεμri)|<<1, riis the radius of the inner conductor) with the parameters of material ε, μ, σ, from which they are made, the formulas of the distributed surface impedance Z¯Sare presented in Table 1.

Design type of vibratorBreadboard view of vibratorFormula for impedance
1The solid metallic cylinder of the rΔ0 radius,Δ0=ω/k2πσωμis the skin-layer thicknessZ¯S=1+iZ0σΔ0
2The dielectrical metalized cylinder with covering, made of the metal of the hR Δ0 thicknessZ¯S=1Z0σhR+ikr(ε1)/2
3The metal-dielectrical cylinder (L1 is the thickness of the metal disk, L2 is the thickness of the dielectric disc)Z¯S=iL2L1+L22krε
4The magnetodielectrical metalized cylinder with the inner conducting cylinder with the radius riZ¯S=1Z0σhRi/krμln(r/ri)
5The metallic cylinder with covering, made of magnetodielectric of the r-ri thickness, or the corrugated cylinderZ¯S=ikrμln(r/ri)
Z¯S(s)=R¯S(s)+iX¯S(s)
6The metallic monofilar helix of the r radius (kr1) with the ψ winding angle
Z¯S=(i/2)krctg2ψ

Table 1.

The formulas of the distributed surface impedance Z¯S

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 μ1/ε1in all formulas for the vibrator in the material medium with the ε1and μ1parameters. We note, that most of the formulas for impedances include the parameters εand μ, smooth change of which (in the case of their dependence from the static electrical and magnetic fields) and the characteristics of radiation of the system, correspondingly, (at its fixed geometrical sizes) can be made, for example, by external field effects.

The constants A¯(±L)and B¯(±L)can be found employing the boundary conditions (7) and the symmetry conditions [5], unambiguously related to a method of vibrator excitation; if E0s(s)=E0ss(s), J(s)=J(s)=Js(s)and A¯(L)=A¯(+L), B¯(L)=B¯(+L); if E0s(s)=E0sa(s), J(s)=J(s)=Ja(s)and A¯(L)=A¯(+L), B¯(L)=B¯(+L). Then, in terms of symmetric and antisymmetric current components, marked by indexes sand a, respectively, for arbitrary vibrator excitation by E0s(s)=E0ss(s)+E0sa(s)it is not difficult to show that

J(s)=Js(s)+Ja(s)=αiωk{LsE0s(s)sink˜(ss)dssink˜(L+s)+αPs[kr,k˜(L+s)]sin2k˜L+αPs(kr,2k˜L)LLE0ss(s)sink˜(Ls)dssink˜(L+s)+αPa[kr,k˜(L+s)]sin2k˜L+αPa(kr,2k˜L)LLE0sa(s)sink˜(Ls)ds},E27

where Psand Paare the functions of vibrator self-fields equal to

Ps[kr,k˜(L+s)]=Ls[eikR(s,L)R(s,L)+eikR(s,L)R(s,L)]sink˜(ss)ds|s=L=Ps(kr,2k˜L),(a)Pa[kr,k˜(L+s)]=Ls[eikR(s,L)R(s,L)eikR(s,L)R(s,L)]sink˜(ss)ds|s=L=Pa(kr,2k˜L).(b)E28

It is evident that if an impedance vibrator is located in restricted volume V, the expression for the current coincides with (27), but the functions of vibrator self-field (28) must contain components of electric Green’s function for corresponding electrodynamic volume.

Let us consider a problem of vibrator excitation at its geometric center by a lumped EMF with amplitude V0. The mathematical model of excitation is presented as

E0s(s)=E0ss(s)=V0δ(s0),E29

where δ(s0)=δ(s)is Dirac delta-function. Then the expression for the current (27) is

J(s)=αV0(iω2k˜)sink˜(L|s|)+αPδs(kr,k˜s)cosk˜L+αPLs(kr,k˜L).E30

Here Pδs(kr,k˜s)=Ps[kr,k˜(L+s)](sink˜s+sink˜|s|)PLs(kr,k˜L)and Ps[kr,k˜(L+s)]are defined by the formula (28a). Explicit expressions for Pδs(kr,k˜s)and PLs(kr,k˜L)can be expressed explicitly in terms of generalized integral functions [4,5]. Thus, PLs(kr,k˜L)which will be needed below may be presented as

PLs(kr,k˜L)=cosk˜L{2ln2γ(L)(1/2)[Cin(2k˜L+2kL)+Cin(2k˜L2kL)](i/2)[Si(2k˜L+2kL)Si(2k˜L2kL)]}+sink˜L{(1/2)[Si(2k˜L+2kL)+Si(2k˜L2kL)](i/2)[Cin(2k˜L+2kL)Cin(2k˜L2kL)]},E31

where Si(x)and Cin(x)are sine and cosine integrals of complex argument.

Since the current distribution (30) is now known we can calculate electrodynamic characteristics of an impedance vibrator. Thus, an input impedance Zin=Rin+iXinof vibrator in a feed point is equal

Zin[Ohm]=V0J(0)=(60ik˜αk)cosk˜L+αPLs(kr,k˜L)sink˜L+αPδL(kr,k˜L),E32

where

PδL(kr,k˜L)=LLeikR(s,L)R(s,L)sink˜|s|ds=sink˜L{γ(L)+(1/2)[Cin(2k˜L+2kL)Cin(2k˜L2kL)]Cin(k˜L+kL)+Cin(k˜LkL)+(i/2)[Si(2k˜L+2kL)Si(2k˜L2kL)]i[Si(k˜L+kL)Si(k˜LkL)]}+cosk˜L{(1/2)[Si(2k˜L+2kL)+Si(2k˜L2kL)]Si(k˜L+kL)Si(k˜LkL)(i/2)[Cin(2k˜L+2kL)+Cin(2k˜L2kL)]+i[Cin(k˜L+kL)+Cin(k˜LkL)]}.E33

Note, that an input admittance Yin=Gin+iBincan be calculated as Yin=1/Zin.

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 ri=0.3175cm), covered by dielectric (ε=9.0)shell (radius r=0.635cm), the experimental data [21] at Figure 2 and 2) metal wire (ri=0.5175cm), covered with ferrite (μ=4.7)shell (r=0.6cm), the experimental data from [22] at Figure 3. The plots show that trends of the theoretical curves coincide with that of the experimental curves, especially near the resonance for Bin=0, though in absolute values some difference is observed. In our opinion, the discrepancy of theoretical curves, obtained by solving the integral equation for the current by averaging method, and the experimental curves may be caused by evident fact that vibrator self-field (19) was averaged and the current amplitude was determined with some error.

Figure 2.

The input admittance of metal wire (radius ri =0.3175 cm), covered by dielectric shell (ε =9.0, radius r =0.635 cm) versus its electrical length at the frequency f =600 MHz

Figure 3.

The input admittance of metal wire (radius ri =5175 cm), covered with ferrite shell (μ =4.7, r =0.6 cm) versus frequency for L =30.0 cm

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 (ε1=μ1=ε2=μ2=1)rectangular waveguide so that its slot has arbitrary orientation in the plane of waveguide cross-section and has no contacts with waveguide walls (Figure 4).

Figure 4.

The problem geometry and notations

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

(d2ds2+k2)LLJ(s)4eikR(s,s)R(s,s)ds=iωH0s(s)(d2ds2+k2)LLJ(s)[G0sV1(s,s)+G0sV2(s,s)]ds,E34

where 4eikR(s,s)R(s,s)is the Green’s function of the slot in infinite perfectly conducting plane, G0sV1,2(s,s)are the Green’s functions, which takes into account multiple reflection from walls of volumes.

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

d2J(s)ds2+k2J(s)=α{iωH0s(s)+F[s,J(s)]+F0[s,J(s)]}.E35

Here α=1/8ln[de/(8L)]is the natural small parameter of the problem (|α|<<1), de=deπh2dis equivalent slot width which takes into account a real wall thickness h(h/λ<<1) [3],

F[s,J(s)]=4dJ(s)dseikR(s,s)R(s,s)+4[d2J(s)ds2+k2J(s)]γ(s)+4LL{[d2J(s)ds2+k2J(s)]eikR(s,s)[d2J(s)ds2+k2J(s)]R(s,s)}dsE36

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

F0[s,J(s)]=dJ(s)ds[G0sV1(s,s)+G0sV2(s,s)]|LL+LL[d2J(s)ds2+k2J(s)][G0sV1(s,s)+G0sV2(s,s)]dsE37

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 A(s)and B(s)which is equivalent to initial equation (35)

dA(s)ds=αk{iωH0s(s)+FN[s,A(s),dA(s)ds,B(s),dB(s)ds]}sinks,dB(s)ds=+αk{iωH0s(s)+FN[s,A(s),dA(s)ds,B(s),dB(s)ds]}cosks,E38

where FN=F+F0is the total self-field of the slot.

Assuming, as in Section 4, dA(s)ds=0and dB(s)ds=0in the right-hand members of equations (38) and making partial averaging over the variable s, we derive the equations of the first approximation by averaging method

dA¯(s)ds=α{iωkH0s(s)+F¯N[s,A¯,B¯]}sinks,dB¯(s)ds=+α{iωkH0s(s)+F¯N[s,A¯,B¯]}cosks,E39

where

F¯N[s,A¯,B¯]=[A¯(s)sinksB¯(s)cosks]GsΣ(s,s)|LL,GsΣ(s,s)=GsV1(s,s)+GsV2(s,s)E40

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

J(s)=A¯(L)cosks+B¯(L)sinks+αLs{iωkH0s(s)+F¯N[s,A¯,B¯]}sink(ss)ds.E41

To determine constants A¯(±L)and B¯(±L)we will use the boundary conditions (7) and the symmetry conditions, uniquely related both to slot excitation method and its position in waveguide. Then, in terms of symmetric and antisymmetric magnetic current components, marked by indexes sand a, respectively, for arbitrary slot excitation by H0s(s)=H0ss(s)+H0sa(s)with an accuracy of order α2we have

J(s)=Js(s)+Ja(s)=αiωk{LsH0s(s)sink(ss)dssink(L+s)LLH0ss(s)sink(Ls)dssin2kL+αNs(kde,2kL)sink(L+s)LLH0sa(s)sink(Ls)dssin2kL+αNa(kde,2kL)},E42

where Ns(kde,2kL)and Na(kde,2kL)are the functions of self-field which are equal

Ns(kde,2kL)=LL[GsΣ(s,L)+GsΣ(s,L)]sink(Ls)ds,Na(kde,2kL)=LL[GsΣ(s,L)GsΣ(s,L)]sink(Ls)ds,E43

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

Supposing that dominant wave H10with amplitude H0is propagated from the region z=, we have

H0s(s)=2H0cosφ[sinπx0acosπ(scosφ)a+cosπx0asinπ(scosφ)a].E44

The symmetric and antisymmetric components of the slot current, relative to the slot center s=0, become equal

J(s)=J0f(s)=α2H0cosφ2iω/k2[1(kφ/k)2][sin2kL+α2Wφsa(kde,2kL)]×{sinπx0asinkL(coskscoskφLcoskLcoskφs)+cosπx0acoskL(sinkssinkφLsinkLsinkφs)},E45

where J0is current amplitude, f(s)is the current distribution function, kφ=πacosφ, Wφsa(kde,2kL)is the function of slot self-field, defined by formulas (43).

Reflection and transmission coefficients, S11and S12for the dominant wave in the slot iris are define by the current as

S11=(1+S12)e2iγz,S12=α16πkgcos2φf(kφL)iabk3[1(kφ/k)2][coskL+α2Wφ(kde,kL)],E46
f(kφL)=2coskφLsinkLcoskφL(kφ/k)coskLsinkφL1(kφ/k)2coskLsin2kφL+2kφL2(kφ/k),BB2

where kg=k2(π/a)2is the propagation constant of H10wave.

Figure 5 shows the theoretical and experimental wavelength dependences of power reflection coefficient |S11|2for the iris, which oriented so that the angle between slot axis {0s}and waveguide axis {0x}are 0Oand 30O.

Figure 5.

Power reflection coefficient |S11|2 versus wavelength for the iris at a =23.0 mm, b =10.0 mm, 2L =16.0 mm, d =1.5 mm, h =2.0 mm, x0=a/2, y0=b/2

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.

Figure 6.

The geometry of three-element vibrator-slot system and notations

Let a fundamental wave H10propagates from the area z=in a hollow infinite rectangular waveguide, the area index is “Wg”. Two thin nonsymmetrical vibrators (monopoles) with variable surface impedance are located in a waveguide with cross-section {a×b}. A narrow transverse slot cut in a broad wall of the waveguide symmetrically relative to its longitudinal axis is radiating into free half-space, the area index is “Hs”. The vibrators radiuses and lengths are r1,2and L1,2((r1,2/L1,2)<<1), the slot width is d, the slot length is 2L3((d/L3)<<1) and the waveguide wall thickness is h. One vibrator is located in the plane {x0y}and the second vibrator may be shifted along the axis {0z}at the distance z0(Figure 6).

For this configuration the system of integral equations relative to electrical currents at the vibrators J1,2(s1,2)and equivalent magnetic current in the slot J3(s3)in accordance with (8) may be represented as

(d2ds12+k2){L1L1J1(s1)Gs1Wg(s1,s1)ds1+L2L2J2(s2)Gs2Wg(s1,s2)ds2}ikL3L3J3(s3)G˜s3Wg(s1,s3)ds3=iω[E0s1(s1)zi1(s1)J1(s1)],(a)(d2ds22+k2){L2L2J2(s2)Gs2Wg(s2,s2)ds2+L1L1J1(s1)Gs1Wg(s2,s1)ds1}==iω[E0s2(s2)zi2(s2)J2(s2)],(b)(d2ds32+k12)L3L3J3(s3)[Gs3Wg(s3,s3)+Gs3Hs(s3,s3)]ds3ikL1L1J1(s1)G˜s1Wg(s3,s1)ds1=iωH0s3(s3).(c)E47

Here Gs1,2Wg(s1,2,s1,2)and Gs3Wg,Hs(s3,s3)are components of the Green’s functions of the rectangular waveguide and the half-space over the plane [3,4], G˜s1Wg(s3,s1)=zGs1Wg[x(s3),0,z;x(s1),y(s1),z0]and G˜s3Wg(s1,s3)=zGs3Wg[x(s1),y(s1),z;x(s3),0,0]after substitution z=0into G˜s1Wgand z=z0into G˜s3Wgafter first derivation, zi1,2(s1,2)is the internal impedance per unit length of the vibrators ([Ohm/m]), E0s1,2(s1,2)and H0s3(s3)are projections of impressed sources fields on the vibrators and the slot axes, s1=L1and s2=L2are end coordinates of mirror vibrator images relative to the lower broad wall of the waveguide [4]

We will seek the solution of equations system (47) by a generalized method of induced EMMF [19,20], using functions J1(2)(s1(2))=J1(2)0f1(2)(s1(2))and J3(s3)=J30f3(s3)as approximating expressions for the currents. Here J1(2)0and J30are unknown current amplitudes, f1(2)(s1(2))and f3(s3)are predetermined functions of the current distributions. In accordance with (27) and (42) for the vibrator-slot structure excited by the fundamental wave H10we have f1(2)(s1(2))=cosk˜1(2)s1(2)cosk˜1(2)L1(2), f3(s3)=cosks3coskL3, k˜1(2)=ki2πzi1(2)avZ0Ω1(2), zi1(2)av=1L1(2)0L1(2)zi1(2)(s1(2))ds1(2)are average values [4] of internal impedances, Ω1(2)=2ln(2L1(2)/r1(2)).

In accordance with the generalized method of induced EMMF, we multiply equation (47a) by the function f1(s1), equation (47b) by the function f2(s2), and the equation (47c) by the function f3(s3)and integrate the equations (47a) and (47b) over the length of the vibrators, and the equation (47c) over the length of the slot. As a result, we obtain a system of linear algebraic equations relative to the current amplitudes J1,2,30

J10Z11Σ+J20Z12+J30Z13=iω2kL1L1f1(s1)E0s1(s1)ds1,J20Z22Σ+J10Z21=iω2kL2L2f2(s2)E0s2(s2)ds2,J30Z33Σ+J10Z31=iω2kL3L3f3(s3)H0s3(s3)ds3.E48

Here

Z11(22)=4πabm=1n=0{εn(k2ky2)k˜1(2)2kkz(k˜1(2)2ky2)2ekzr1(2)sin2kxx01(02)×[sink˜1(2)L1(2)coskyL1(2)k˜1(2)kycosk˜1(2)L1(2)sinkyL1(2)]2},E49
F1(2)z=ir1(2)0L1(2)f1(2)2(s1(2))Z¯S1(2)(s1(2))ds1(2),E50
Z12=Z21=4πabm=1n=0{εn(k2ky2)k˜1k˜2ekzz0kkz(k˜12ky2)(k˜22ky2)sinkxx01×sinkxx02[sink˜1L1coskyL1(k˜1/ky)cosk˜1L1sinkyL1]×[sink˜2L2coskyL2(k˜2/ky)cosk˜2L2sinkyL2]},E51
Z33Hs=Si4kL3iCin4kL32coskL3[2(sinkL3kL3coskL3)×(ln16L3deCin2kL3iSi2kL3)+sin2kL3eikL3],E52
Z33Wg=8πabm=1,3..n=0,1..{εnkkz(k2kx2)ekzde4[sinkL3coskxL3(k/kx)coskL3sinkxL3]2},E53
Z13=Z31==4πabm=1n=0{εnkk˜1ekzz0i(k˜12ky2)(k2kx2)sinkxx01sinkxa2[sink˜1L1coskyL1k˜1kycosk˜1L1sinkyL1]×[sinkL3coskxL3kkxcoskL3sinkxL3]},Z11(22)Σ=Z11(22)+F1(2)z,Z33Σ=Z33Hs+Z33Wg,E54

where εn={1,n=02,n0, kx(y)=m(n)πa(b), kz=kx2+ky2k2, m, nare integers; Siand Cinare integral sine and cosine.

The energy characteristics of the vibrator-slot system: the reflection and transmission coefficients, S11and S12, and power radiation coefficient |SΣ|2, are defined by the expressions

S11=4πiabkkg{J32kg2k2f(kL3)J1kgk˜1sin(πx01a)f(k˜1L1)eikgz0J2kgk˜2sin(πx02a)f(k˜2L2)}e2ikgz,E55
S12=1+4πiabkkg{J32kg2k2f(kL3)+J1kgk˜1sin(πx01a)f(k˜1L1)eikgz0+J2kgk˜2sin(πx02a)f(k˜2L2)},E56
|SΣ|2=1|S11|2|S12|2E57

In expressions (55)-(57)

J1=1/(Z11ΣZ22ΣZ33ΣZ21Z12Z33ΣZ31Z13Z22Σ)×[k2kgk˜1sinπx01af1(k˜1L1)eikgz0Z22ΣZ33Σk2kgk˜2sinπx02af2(k˜2L2)Z12Z33Σf3(kL3)Z13Z22Σ],BB3
J2=1/(Z11ΣZ22ΣZ33ΣZ21Z12Z33ΣZ31Z13Z22Σ)×[k2kgk˜2sinπx02af2(k˜2L2)(Z11ΣZ33ΣZ31Z13)k2kgk˜1sinπx01af1(k˜1L1)eikgz0Z21Z33Σ+f3(kL3)Z13Z21],BB4
J3=1/(Z11ΣZ22ΣZ33ΣZ21Z12Z33ΣZ31Z13Z22Σ)×[f3(kL3)(Z11ΣZ22ΣZ21Z12)+k2kgk˜2sinπx02af2(k˜2L2)Z12Z31k2kgk˜1sinπx01af1(k˜1L1)eikgz0Z31Z22Σ],BB5
f1(2)(k˜1(2)L1(2))=sink˜1(2)L1(2)k˜1(2)L1(2)cosk˜1(2)L1(2),BB6
f3(kL3)=sinkL3cos(πL3/a)(ka/π)coskL3sin(πL3/a)1[π/(ka)]2.BB7

Let us consider several distribution functions for the surface impedance along the vibrator, namely: 1) ϕ0(s1(2))=1, the constant distribution, 2) ϕ1(s1(2))=2[1(s1(2)/L1(2))], the triangular distribution linear decreasing to the vibrator end, and 3) ϕ2(s1(2))=2(s1(2)/L1(2)), the triangular linear increasing distribution. All distribution have equal average values ϕ0,1,2(s1(2))¯=1. The expression for F1(2)z0with the distribution function 1), in accordance with (50), can be presented as

F1(2)z0=2i(R¯S1(2)+iX¯S1(2))k˜1(2)2L1(2)r1(2)[(k˜1(2)L1(2)2)2(2+cos2k˜1(2)L1(2))38k˜1(2)L1(2)sin2k˜1(2)L1(2)]=F˜1(2)z(R¯S1(2)+iX¯S1(2))Φ1(2)E58

with the distribution function 2) as

F1(2)z1=F˜1(2)z×{R¯S1(2)Φ1(2)+iX¯S1(2)[(k˜1(2)L1(2)2)2(2+cos2k˜1(2)L1(2))74sin2k˜1(2)L1(2)2(cosk˜1(2)L1(2)1)]}E59

and with the distribution function 3) as

F1(2)z2=F˜1(2)z{R¯S1(2)Φ1(2)+iX¯S1(2)[(k˜1(2)L1(2)2)2(2+cos2k˜1(2)L1(2))+74sin2k˜1(2)L1(2)34k˜1(2)L1(2)sin2k˜1(2)L1(2)+2(cosk˜1(2)L1(2)1)]}E60

Since the formulas for F1(2)z0,1,2differ from one another, in spite of equal average values of functions ϕ0,1,2(s1(2))and identical functional dependences in formulas for currents, the current amplitudes and, hence, energy characteristics will be substantially different.

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: a=58.0 mm, b=25.0 mm, h=0.5 mm, r1,2=2.0 mm, L1,2=15.0 mm, R¯S1(2)=0, x01=a/8, x02=7a/8, d=4.0 mm and 2L3=40.0 mm.

Figure 7.

The energy characteristics versus wavelength at λv1,2res=λslres, Z¯S1=Z¯S2

Figure 8.

The energy characteristics versus wavelength at λv1,2res≠λslres, Z¯S1≠Z¯S2, experimental data are marked by circles

The choice of slot dimensions was stipulated by its natural resonance at the average wavelength of the waveguide frequency range λ3res=86.0 mm. The dimensions of the vibrators have been selected so that their resonant wavelength was within the waveguide operating range. Here we present the results only for vibrators with inductive impedances (X¯S1(2)>0), known to increase the vibrator electrical length, i.e. to increase λ1,2resas compared to case Z¯S1(2)=0, without decreasing a distance between the vibrators ends and the upper broad wall of the waveguide. This is very important for increasing the breakdown power for waveguide device as a whole.

As might be expected from physical considerations, displacement of the impedance vibrator along the longitudinal axis of the waveguide at a distance z0from the centre of the slot, where the maximum mutual influence between elements of the structure is observed, are multiple of λG/4(Fig. 7: z0=λG/4=32.0 mm and z0=λG/2=64.0 mm). Here λG=2π/(2π/λslres)2(π/a)2is resonant wavelength of the slot in the waveguide, and λslresis the resonant wavelength of the slot in the free half-space over the plane. As seen from Figure 7, an acceptable reflection coefficient |S11|and high level of radiation could not be achieved if the monopoles have the equal distributed impedances Z¯S1=Z¯S2. The maximum of radiation coefficient |SΣ|2and almost perfect agreement with the feed line, as well as tuning to other resonant wavelengths can be achieved by changing the distribution functions of impedance along the monopoles axes (Figure 8). Fig. 8 also shows that the results of mathematical modeling are confirmed by the experimental data. Experimental models have been made in the form of corrugated brass rods (see photo in Figure 8).

For the arbitrary vibrator-slot structures and coupled electrodynamic volumes expressions for fvs,a(sv)and fsls,a(ssl)(the subscripts s,adenote the symmetric and antisymmetric components of the currents with respect to the vibrator (sv=0) and slot (ssl=0) centers, respectively), in accordance with the results, presented in Sections 4 and 5 (see formulas (27) and (42)), can be obtained from the following relations

fvs,a(sv){sink˜(Lvsv)LvsvE0svs,a(sv)sink˜(Lv+sv)dsv+sink˜(Lv+sv)svLvE0svs,a(sv)sink˜(Lvsv)dsv},(a)fsls,a(ssl){sink(Lslssl)LslsslH0ssls,a(ssl)sink(Lsl+ssl)dssl+sink(Lsl+ssl)sslLslH0ssls,a(ssl)sink(Lslssl)dssl},(b)E61

where E0svs,a(sv)and H0ssls,a(ssl)are projections of symmetric and antisymmetric components of impressed sources on the vibrator and the slot axes. Here the sign ~ means that after integration in expressions (61) only multipliers, depending upon coordinates svand ssl, are left.

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 {0ssl}and {0z}coincide, we obtain

fsls(ssl)=cosksslcoskgLslcoskLslcoskgssl,fsla(ssl)=sinksslsinkgLslsinkLslsinkgssl.E62

If vibrator is excited at its base by voltage δ-generator as in a waveguide-to-coaxial adapter we have

fv(sv)=sink˜(Lvsv).E63

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.

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Mikhail V. Nesterenko, Sergey L. Berdnik, Victor A. Katrich and Yuriy M. Penkin (November 18th 2015). Electromagnetic Waves Excitation by Thin Impedance Vibrators and Narrow Slots in Electrodynamic Volumes, Advanced Electromagnetic Waves, Saad Osman Bashir, IntechOpen, DOI: 10.5772/61188. Available from:

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Generalized Orthogonality Relation for Spherical Harmonics

By A. Draux and G. Gouesbet

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