Application of Generalized Equivalent Circuits (GEC) Method Based on the Polarization Current, to Analyses Electromagnetic Scattering by Dielectric Slab with Neglected Thickness: Application to the Characterization of Vegetation Leaves KRRAOUI Houssemeddine

AGUILI Taoufik

SYSCOM Laboratory National Engineering School of Tunis, ENIT Tunis, Tunisia [email protected]

SYSCOM Laboratory National Engineering School of Tunis, ENIT Tunis, Tunisia [email protected]

Abstract—In this paper, we develop an electromagnetic analysis of a rectangular waveguide loaded with a dielectric model that is composed of a leaf plant with a neglected thickness. One numerical method is used for modeling the proposed structures: the moment method combined with Generalized Equivalent Circuit (MoM-GEC). We can determine the wave diffraction according to the dielectric properties. Our obtained electromagnetic field verified the boundary condition.

leaves. This leaf is parallel venation one with a midrib more developed than the minor veins Figure.1.

I. I NTRODUCTION The water content is a very important parameter for many industrial processes, especially for the agribusiness and the building materials applications. In fact, knowledge of medium water content can, in advance, determine the precocity of the agriculture seeds varieties, improve the plants storage and restrict the energy expenses on drying operations. The traditional measurement techniques, such as the chemical processes [1] or the gravimetric method, are destructive and do not allow a continuous assessment, it would be interesting to have a measuring instrument that permit to evaluate the water contents and their variations in a continuous way. The microwave techniques are based on studying the interaction of electromagnetic waves in microwave range and materials under test [2]. Therefore, they are nondestructive and instantaneous. These latter microwave techniques require a direct and inverse problems analysis. The study of the direct problem allows to calculate the reflection coefficient following the dielectric properties of the material under test. The determination of the permittivity from the measured reflection coefficient defines the inverse problem. Water inside leaves is generally distributed between different tissues in not well known proportions. To understand and predict its dielectric behavior, it is necessary to involve its morphology. This study is done exclusively on corn

Figure 1.

The various compartments to be characterized of a corn leaf.

The Veins, being richer in water than the blade, it is thus possible to focus on a study according to a model where dielectric rectangular layers with neglected thickness compared with the wavelength are inserted into a waveguide. In this paper, we present a new electromagnetic analysis that aim to study the direct problem. Here, our work can illustrate an electromagnetic study that usually use an Integral Equation (IE) Method combined with the Generalized Equivalent Circuit (GEC) [3]. This method is more adapted to carry out an electromagnetic study of the microwaves planar structures. Indeed, this method allows to reduce the dimension of the problem under consideration that we can write the initial boundary conditions following the integral equation form witch definite on the obstacles surface. When the complexity of the studied structures increases and the resolution becomes complicated. The equivalent circuits that are introduced in the development of the Integral Equation methods are useful to transpose the field problems to equivalent circuits problems in order to simplify the associated treatment. The latest method is

proposed by Marcuvitz [4] and generalized by Baudrand [5] in order to solve the Maxwell’s equation. This representation (GEC) is used to express the boundary conditions of the unknown electromagnetic field state with one electrical equivalent circuit. This method allows as taking into account the upper order mode effects. It assures a better precision to measure the dielectric properties of the material need to be characterized. It justifies our choice of this modeling method in a rectangular wave-guide. The relationship between scattering matrix and geometrical and dielectric material parameters of dielectric is obtained.

II. MODEL FOR A DIELECTRIC SLAB The model of the plants leaf inside a waveguide is depicted in fig.2.

Figure 3.

Let’s (fmn ){m,n=(1,2,...M,N )} be the local modal basis of the EEEE waveguide. The excitation source are E0 = V0 .f0 where f0 represent the active mode and V0 the amplitude. b is expressed in (3) as a function The impedance operator Z of the higher-order modes hfmn |, and their modes impedance z1mn XX α α α b= Z |fmn izmn hfmn |,(α=T E,T M) (3) n

Figure 2. Dielectrics obstacle of finite length in a waveguide (a=22.860 mm x b=10.160 mm).

The plan of discontinuity consists of two complementary domains in which the relations of passage change. In the empty − → domain the current density of polarization Jp nullifies. We note − → Ee its dual. In the dielectric domain the relation of passage is translated by a law of ohm: bs .Jp Ee = 2Z

(1)

bs Indicate the impedance of surface associated a dielectric Or Z domain expressed as follow: bs = |Πi Z

1 hΠ| j.ω.(εr − 1).ε0 .δ

(2)

(δ

AGUILI Taoufik

SYSCOM Laboratory National Engineering School of Tunis, ENIT Tunis, Tunisia [email protected]

SYSCOM Laboratory National Engineering School of Tunis, ENIT Tunis, Tunisia [email protected]

Abstract—In this paper, we develop an electromagnetic analysis of a rectangular waveguide loaded with a dielectric model that is composed of a leaf plant with a neglected thickness. One numerical method is used for modeling the proposed structures: the moment method combined with Generalized Equivalent Circuit (MoM-GEC). We can determine the wave diffraction according to the dielectric properties. Our obtained electromagnetic field verified the boundary condition.

leaves. This leaf is parallel venation one with a midrib more developed than the minor veins Figure.1.

I. I NTRODUCTION The water content is a very important parameter for many industrial processes, especially for the agribusiness and the building materials applications. In fact, knowledge of medium water content can, in advance, determine the precocity of the agriculture seeds varieties, improve the plants storage and restrict the energy expenses on drying operations. The traditional measurement techniques, such as the chemical processes [1] or the gravimetric method, are destructive and do not allow a continuous assessment, it would be interesting to have a measuring instrument that permit to evaluate the water contents and their variations in a continuous way. The microwave techniques are based on studying the interaction of electromagnetic waves in microwave range and materials under test [2]. Therefore, they are nondestructive and instantaneous. These latter microwave techniques require a direct and inverse problems analysis. The study of the direct problem allows to calculate the reflection coefficient following the dielectric properties of the material under test. The determination of the permittivity from the measured reflection coefficient defines the inverse problem. Water inside leaves is generally distributed between different tissues in not well known proportions. To understand and predict its dielectric behavior, it is necessary to involve its morphology. This study is done exclusively on corn

Figure 1.

The various compartments to be characterized of a corn leaf.

The Veins, being richer in water than the blade, it is thus possible to focus on a study according to a model where dielectric rectangular layers with neglected thickness compared with the wavelength are inserted into a waveguide. In this paper, we present a new electromagnetic analysis that aim to study the direct problem. Here, our work can illustrate an electromagnetic study that usually use an Integral Equation (IE) Method combined with the Generalized Equivalent Circuit (GEC) [3]. This method is more adapted to carry out an electromagnetic study of the microwaves planar structures. Indeed, this method allows to reduce the dimension of the problem under consideration that we can write the initial boundary conditions following the integral equation form witch definite on the obstacles surface. When the complexity of the studied structures increases and the resolution becomes complicated. The equivalent circuits that are introduced in the development of the Integral Equation methods are useful to transpose the field problems to equivalent circuits problems in order to simplify the associated treatment. The latest method is

proposed by Marcuvitz [4] and generalized by Baudrand [5] in order to solve the Maxwell’s equation. This representation (GEC) is used to express the boundary conditions of the unknown electromagnetic field state with one electrical equivalent circuit. This method allows as taking into account the upper order mode effects. It assures a better precision to measure the dielectric properties of the material need to be characterized. It justifies our choice of this modeling method in a rectangular wave-guide. The relationship between scattering matrix and geometrical and dielectric material parameters of dielectric is obtained.

II. MODEL FOR A DIELECTRIC SLAB The model of the plants leaf inside a waveguide is depicted in fig.2.

Figure 3.

Let’s (fmn ){m,n=(1,2,...M,N )} be the local modal basis of the EEEE waveguide. The excitation source are E0 = V0 .f0 where f0 represent the active mode and V0 the amplitude. b is expressed in (3) as a function The impedance operator Z of the higher-order modes hfmn |, and their modes impedance z1mn XX α α α b= Z |fmn izmn hfmn |,(α=T E,T M) (3) n

Figure 2. Dielectrics obstacle of finite length in a waveguide (a=22.860 mm x b=10.160 mm).

The plan of discontinuity consists of two complementary domains in which the relations of passage change. In the empty − → domain the current density of polarization Jp nullifies. We note − → Ee its dual. In the dielectric domain the relation of passage is translated by a law of ohm: bs .Jp Ee = 2Z

(1)

bs Indicate the impedance of surface associated a dielectric Or Z domain expressed as follow: bs = |Πi Z

1 hΠ| j.ω.(εr − 1).ε0 .δ

(2)

(δ