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)
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with boundary conditions:
and initial conditions:
We note that the functions f(z) and F(z) can have discontinuities at the point z=0:
We seek a solution of the jump problem in the space of continuously dierentiable
functions. Let
be some known functions dened in the
spaces
and .
Earlier in (16), we considered the over-determined boundary problem for
telegraph equation (17) in the right quarter of the plane (z > 0, t > 0) with initial
and boundary conditions (see Figure 2):
Figure 2. Traces of functions of the over-determined boundary problem for telegraph equation in the
upper half-plane t>0.
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We obtained in (16) the condition for the solvability of this problem in the
following form:
or
Note that this equality establishes the relationship between the boundary functions
in a mixed boundary problem for telegraph equation in a quarter of a plane. If
the condition of solvability of the over-determined problem is executed, then it
can be extended to the whole quarter-plane and even to the half-plane.
Now we consider the over-determined problem in the left quarter of the plane (z
< 0, t > 0) with initial and boundary conditions
We replace with and seek the function . The equation
itself does not change, and the initial and boundary conditions take the form
and
Then, in a similar way as in (16), we obtain the solvability condition for this
problem in the following form:
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We use the solvability conditions and reduce solving the jump problem with zero
initial conditions to solving the system of four equations:
We denote . If the solvability conditions are added and subtracted, then we
get two integral equations:
If and is dierent from 0, then the function is
found as a solution for the integral equation:
(18)
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If , then the function , and is found as a
solution for the integral equation:
(19)
Thus, we reduced solving the jump problem for telegraph equation to solving the
Volterra integral equations (18) and (19). Solving the rst equation, we can nd the
longitudinal components in the jump problem for H
z,m
(z,t), which corresponds to
the jump problem for the equation (17) with
in the boundary conditions.
The second equation can be used to determine E
z,m
(z,t) with .
4. BOUNDARY CONDITIONS OF THE JUMP PROBLEM
FOR TELEGRAPH EQUATIONS
We express the right parts in the formulas (12) and (15) through known functions.
From the conditions of the jump problem for the tangential components of the
electric and magnetic vectors (2), (3), we obtain the conjugation conditions for
the normal components of these vectors. We express the terms of the left-hand
sides under conjugation conditions through E
z,m
(z,t) and H
z,m
(z,t). For this, we use
the representations of the transverse eld components in formulas (6), (7), (9),
(10) H
x
(x, y, z, t), H
y
(x, y, z, t), E
x
(x, y, z, t), E
y
(x, y, z, t), we substitute them into the
corresponding conditions (2), (3) and for any
we get:
(20)
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(21)
In the system (20), we dierentiate the rst equation with respect to x, the second
with respect to y and add them. Then we get:
Now let us dierentiate the rst equation with respect to y, the second one with
respect to x, and we subtract the second equation from the rst equation. Then
we obtain:
We perform similar transformations in the system (21) and obtain the following
equations:
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In square brackets, the expressions for the sum of the derivatives of the functions
φ
m
(x, y) and ψ
m
(x, y) represent the Laplace operator applied to these functions,
respectively. We use this property of the eigenfunctions. Next, we scalar multiply
both sides of the equations on φ
k
(x, y) and ψ
k
(x, y), and, using the orthogonality of
these functions, we obtain:
(22)
(23)
We express the time derivatives in the systems of equations (22) and (23) as follows:
We consider that the initial conditions are zero, and we get:
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The last two expressions and the rst equations in the systems of equations (22),
(23) are the conjugation conditions on the waveguide cross section in the jump
problem for H
z,m
(z, t) and E
z,m
(z, t), respectively.
We use the obtained results and write down the jump problem for telegraph
equation with respect to H
z,m
(z, t). We assume that A
x
(x, y, t) = j
x
(x, y, t) and A
y
(x, y,
t) = j
y
(x, y, t), B
x
(x, y, t) = 0 and B
y
(x, y, t) = 0. Then we get:
Since the rst condition is homogeneous, we denote
, and then the new desired function is the solution for the Volterra integral
equation of the second kind (for the function g(h) by the formula (18)
(24)
where .
After calculating the trace of the function
on the waveguide cross section,
solving the jump problem is reduced to the recovery of two functions
and
by the following formulas from [16]:
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The jump problem for E
z,m
(z, t) takes the following form:
In this problem, we have a homogeneous second condition, then
and the limit value of the function on
the cross section of the waveguide we nd as a solution for the Volterra integral
equation of the rst kind by formula (19):
(25)
The values of the functions and in the entire waveguide are
determined by the following two formulas from [16]:
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Thus, all components of the electromagnetic eld can be expressed through
solutions for the Volterra equations (24) and (25).
5. CONCLUSIONS
The problem of the excitation of a cylindrical waveguide by the surface currents
on innitely thin metal plate located in the cross section is considered. The
components of the excited electromagnetic eld in the waveguide are searched
in the form of series in the eigenfunctions of the Laplace operator. The jump
problem for searching the unknown coecients of these series is reduced to
solving a system of the Volterra integral equations.
6. ACKNOWLEDGEMENTS
The work is performed according to the Russian Government Program of
Competitive Growth of Kazan Federal University.
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