REDUCING THE PROBLEM OF WAVEGUIDE

EXCITATION BY CURRENTS IN CROSS-

SECTION TO A SYSTEM OF INTEGRAL

VOLTERRA EQUATIONS

Angelina Markina

Kazan Federal University, Kazan, Russia.

Nikolai Pleshchinskii

Kazan Federal University, Kazan, Russia.

Dmitrii Tumakov

Kazan Federal University, Kazan, Russia.

E-mail: m8angelina@gmail.com

Recepción: 05/08/2019 Aceptación: 09/09/2019 Publicación: 23/10/2019

Citación sugerida:

Markina, A., Pleshchinskii, N. y Tumakov, D. (2019). Reducing the problem of waveguide

excitation by currents in cross-section to a system of integral volterra equations. 3C

TIC. Cuadernos de desarrollo aplicados a las TIC. Edición Especial, Octubre 2019, 106-125. doi:

https://doi.org/10.17993/3ctic.2019.83-2.106-125

Suggested citation:

Markina, A., Pleshchinskii, N. & Tumakov, D. (2019).Reducing the problem of waveguide

excitation by currents in cross-section to a system of integral volterra equations. 3C TIC.

Cuadernos de desarrollo aplicados a las TIC. Special Issue, October 2019, 106-125. doi: https://

doi.org/10.17993/3ctic.2019.83-2.106-125

3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529

108

ABSTRACT

The problem of excitation of a cylindrical metal waveguide by a source

located in the cross section is considered. We assume that the source is surface

currents on a at, innitely thin metal plate with a smooth boundary. The plate

is connected to the generator of non-harmonic oscillations. The boundary of

the cross section of a waveguide lled with a homogeneous dielectric is a closed

piecewise-smooth contour. The initial physical problem is formulated as a mixed

boundary problem for the system of the Maxwell equations. Components of the

desired solution for the problem is presented in the form of a series in two sets

of two-dimensional eigenfunctions of the Laplace operator. The rst set of the

eigenfunctions corresponds to the operator with Dirichlet boundary conditions,

the second set to the operator with Neumann boundary conditions. We show

that the expansion coecients of the longitudinal components (components

directed along the waveguide axis) of the electric and magnetic intensity vectors

must be solutions to the jump problem for a system of telegraph equations. The

problem of nding the unknown coecients of the expansion of the longitudinal

component of the vector of electric intensity is reduced to solving a system of

the Volterra integral equations of the rst kind with respect to the derivatives

of the desired coecients. The unknown coecients of the expansion of the

longitudinal component of the vector of magnetic intensity are found by solving

a system of the Volterra integral equations of the second kind.

KEYWORDS

Metal waveguide, Wave excitation, Telegraph equation, Cross-sectional source,

Volterra equation.

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1. INTRODUCTION

Metal waveguides are widely used in electronics and engineering. The study of

such waveguide structures includes both the description of the set of eigenwaves

and the search for the conditions of their excitation (Barybin, 2007). In particular,

the excitation of oscillatory processes with specied characteristics in such

structures is one of the tasks facing engineers.

In the case of a harmonic non-stationary electromagnetic eld, the fundamentals

of the theory of waveguides with metal walls were created in the middle of the

last century (see, for example, works) (Samarskii & Tikhonov, 1948; Samarskii

& Tikhonov, 1947). The problem of eld excitation by currents given inside the

waveguide was investigated in enough detail. The modern theory of excitation

of waveguides of various types is presented in the review article (Solncev, 2009;

Ghaderi & Mahdavi Panah, 2018). For metal waveguides, there are cases when

solutions to the problems of propagation and diraction of eigenwaves can be

obtained analytically (Collin, 1960; Mittra, 1971).

Various methods are used to excite waveguides. For example, in optical

waveguides, geometric inhomogeneities on a dielectric are often used to excite

oscillations by an incident external wave (Sun & Wu, 2010; Shapochkin et al.,

2017; Kheirabadi & Mirzaei, 2019; Kashisaz & Mobarak, 2018). For metal

waveguides, adjacent transducer waves are used or, more often, probes inside the

waveguide (Yirmiyahu, Niv, Biener, Kleiner, & Hasman, 2007; Kong, 2002; Pan

& Li, 2013; Eslami & Ahmadi, 2019; Jabbari et al., 2019; Nakhaee & Nasrabadi,

2019). In this case, the probes can have both a simple dipole shape and a loop

shape. Also, the natural waves are excited through the slits of the waveguide or

through another conjugate waveguide (Sadiku, 2014). In this case, the waveguide

itself can be both homogeneous and inhomogeneous lling (Bogolyubov et al.,

2003; Islamov et al., 2017; Sailaukyzy et al., 2018).

In the present work, we consider the problem of the excitation of a cylindrical

metal waveguide by currents on an innitely thin metal plate located in cross

section and connected to a generator. We assume that the waveguide cross section

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is bounded by a piecewise smooth curve. The non-harmonic electromagnetic

eld excited in the waveguide is sought as a solution to the jump problem for the

Maxwell equations. We show that the longitudinal components of the eld must

be solutions of the system of telegraph equations. The jump problem for such the

system of equations is reduced to the system of the Volterra integral equations.

2. PROBLEM STATEMENT

Let an innite cylindrical waveguide with metal walls (Figure 1a) is lled with a

homogeneous isotropic dielectric, and the

axis is the longitudinal axis of the

waveguide. Let its cross section Ω

be bounded by a piecewise-smooth

contour C and consists of two parts: M and N (Figure 1b), moreover, Ω

. Part

M is an innitely thin ideally conducting plate connected to a generator of high-

frequency non-harmonic oscillations. The currents arising on the plate excite an

electromagnetic eld in the waveguide. It is necessary to nd this eld.

A B

Figure 1. The construction of the cylindrical waveguide with a current source in a cross section in a

plane z = 0

As is known (Nikolskij & Nikolskaya, 1989), the following necessary conjugation

conditions (boundary conditions) are fullled at the interface: the tangential

component of the electric intensity

is continuous, the jump of the tangential

component of magnetic intensity

is equal to the density of the surface current,

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the jump of the normal component of the electric induction is equal to the

density of the surface charge, the normal component of magnetic induction

is

continuous.

We assume that the eld is non-harmonically dependent on time. We search for

solutions of the Maxwell equations at

and at :

(1)

on the set jumps to Ω of the tangent components of the vectors and :

(2)

(3)

Conditions (2) and (3) are slightly more general than required. In the problem of

excitation of a waveguide by a source at a cross section, the jump in the tangential

component of the magnetic vector is set to M (this is the electric current density)

and is zero at N; the jump of the tangent component of the electric vector is

everywhere zero.

The desired solutions of the Maxwell equations (1) must also satisfy the initial

conditions at

:

(4)

and be suciently smooth at and at . We assume that their limit

values are correctly determined at

in the classical or generalized sense

(Pleshchinskii, 2019).

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3. JUMP PROBLEM FOR TELEGRAPH EQUATIONS

Let us proceed from the jump problem for the Maxwell equations (1)-(4) in the

cylindrical domain Ω

R to the jump problem for an innite system of telegraph

equations. The components of the solutions for the Maxwell equations in a

cylindrical region that satisfy the boundary conditions on the wall of the waveguide

(the tangent component of the electric vector is zero) can be represented for the

components H(x, y, z, t) in the following form (Pleshchinskii et al., 2017):

(5)

(6)

(7)

and for components E(x, y, z, t) as follows:

(8)

(9)

(10)

where ε is the dielectric constant, and μ is the magnetic permeability of

the substance. The functions

and are orthonormal sets of

eigenfunctions for the Laplace operator in the domain Ω, with Neumann and

Dirichlet boundary conditions, respectively. Moreover,

and are eigenvalues

of the Laplace operator. We assume that a piecewise-smooth contour is such

that the eigenfunctions exist in simple cases; and in simple cases, as a circle or a

rectangle, they are constructed analytically, in other cases, they are constructed

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numerically. The expansion coecients and are new unknown

functions and must be solutions to jump problems for telegraph equations

:

(11)

(12)

(13)

and for telegraph equations :

(14)

(15)

(16)

where is the wave number ( ). The functions , ,

, are some known functions in domains Ω. We show below

what these functions will be equal to. The functions are dened on M as functions

of the current density, and on N they are equal to zero.

Now we show that the desired expansion coecients for the longitudinal eld

components in the problems (11)-(13) and (14)-(16) can be found after solutions

for some integral equations are obtained.

For this purpose, we consider the jump problem for telegraph equation in the

general case with respect to the auxiliary function u(z, t) in the half-plane t > 0:

(17)